Were they? Really? That is the question one might has to ask before anything else.

And the easy part on this question is simply that they were not spared, at all. Not in the first wave and especially not in later waves.

This map in question is a distortion of known reality, based on one influential map from 1962 (discussed further down).
The apparent impression that Milan would be have been 'spared' is based on a contemporary account of a Florentine plague survivor, of dubious reliability, and providing only very limited evidence over any relationship between possible cause and effect, as question-map and the witness-account rest on sparse data and biases:

Piacenza was fewer than 42 miles south of Milan, a city of nearly 100,000 people. And yet secure contemporary evidence of the Black Death in Lombardy is almost non-existent (Albini 1982: 14–17). Why would Milan and many other towns and cities to the north of the Po River, all of which were larger than most rural market centers anywhere in Europe and Britain, be spared a disease supposedly spread by contagion? Oddly, many historians accept the claim by a Florentine plague survivor, Matteo Villani, that Milan escaped catastrophic mortality in 1348–49 because its ruler took cruel and aggressive isolationist measures to board up infected houses when the first cases of the new disease appeared. While a few places in Milanese suburban districts reported epidemic outbreaks in 1350, during the high traffic caused by pilgrims to Rome this Jubilee year, no evidence of these outbreaks is unambiguously plague-related.

Instead, the first devastating plague in Milan and the major cities of Lombardy occurred in 1361–63 (Del Panta 1982: 118). Francesco Petrarca fled Milan in the late spring of 1361, as the city faced its initial experience with the catastrophic new epidemic. His son remained behind. Long disappointed by the young man’s adolescent choices and limited achievements, Petrarca consigned his nuanced remorse to pages of a precious manuscript copy of Virgil which his own father had given him:

Our Giovanni,[…] died in the year of our Lord 1361, […] He died in Milan in the unexampled general devastation wrought by the plague, which hitherto had left that city immune from such evils, but now has found it and has invaded it.

Many of Petrarca’s other correspondents survived the initial wave of plague, only to die in the next epidemic. Francesco Nelli subsequently died of plague, in Avignon, in 1363.
_{— Ann G. Carmichael: "Plague Persistence in Western Europe: A Hypothesis", 'Pandemic Disease in the Medieval World: Rethinking the Black Death', The Medieval Globe, Vol. 1, No. 1, Article 8, 2014. (link)}

As did the initial hypothesis generating historian Matteo Villani, in 1363.

We cannot see as much as we'd like from this map. We see a mere observation and should be quite cautious in deducing from such a correlation any form of causation. Post hoc ergo propter hoc is a fallacy.

Further, if we see you see a mainstream newspaper opinion piece dictated in 2020, and thus mentioning 'corona' the level of wariness should increase to extreme levels, as the amount of imprecision and propaganda reached new heights at the cost of scientific accuracy in that year. If you read for example a Washington Post article about the Black Death of 1347–1351 in Milan which then tells tales about measures being supposedly effective then, despite that they were only introduced after 1370s, that should ring a bell or twice.

The above map depicts a temporal aggregate over a few years. As such it also does not mark areas which in 1347/8 were not affected, at all (?), 'measures' taken or not, despite being surrounded by areas which were hit hard. Frankonia was so 'spared' by the Plague in 1347/8 for example but hit much harder afterwards, just as Milan itself.

What remarkable rats they were! To have crossed the sea and to have marched into remote English villages, and yet to have effectively bypassed the cities of Milan, Liege and Nuremberg,” where the incidence of plague was very low. (Milan, it may be noted, enforced a quarantine that may have saved its citizens from the plague.)
_{— Norman F Cantor: "In the Wake of the Plague. The Black Death and the World it Made", Free Press: New York, London, 2001.}

For Bruges the effects were quite nasty on the development of the city, approaching apparently 31% mortality – which was previously just assumed as 'being low' from a mere laziness of copying and pasting unreferenced claims made up from argumentum ex silentium.
(— Jan Vandeburie: "De Zwarte Dood te Brugge. Een status questionis en enkele nieuwe beschouwingen’, Handelingen van het Genootschap voor Geschiedenis te Brugge, 147/2 (2010), 269-308.
— Joris Roosen: "Severity and Selectivity of the Black Death and Recurring Plague in the Southern Netherlands (1349-1450), TSEG 14 (4): 25-55 doi: 10.18352/tseg.986)

)

Milan was hit with plague in 1348 – but indeed had only few deaths then:

Diagnosis Cities hardest hit during the outbreak of 1348-1350 tried to take measures to control an epidemic no one understood. "In Milan, to take one of the most successful examples, city officials immediately walled up houses found to have the plague, isolating the healthy in them along with the sick. Venice took sophisticated and stringent quarantine and health measures, including isolating all incoming ships on a separate island. But people died anyway, though fewer in Milan and Venice than in cities that took no such measures"
_{— EL Knox: "The Black Death", 1995, p9 (cited after The Bubonic Plague (Yersinia pestis): "The Black Death").}

What did they do, exactly?

Milan began the practice of forced isolation when a few plague cases first appeared in 1348. Headed by a lord, Bernabò Visconti, rather than an elected council, the Milanese government could act swiftly and ruthlessly. They sealed off the first few victims and their families in their houses until all had died or the survivors had proved their good health.
Agnolo di Tura reported deaths in only three families, and so Milan was spared the horrors suffered by so many other Italian cities. But Milan was not to repeat its success during subsequent plagues, nor did other Italian cities follow suit until the later fifteenth century.

Whether discouraged by the theory of corrupted air, the lack of organizational resources, or the assault on human dignity that shutting in entailed, few later medieval cities practiced this method. For the most part the governments that pioneered the policy—Visconti Milan, grand ducal Florence, and Elizabethan England—were well-organized and authoritarian governments. In many ways this behavior echoed a new and ruthless tone in political theory and practice.
_{— Jospeh P Byrne: "Daily Life during the Black Death", Daily Life Through History, The Greenwood Press: Westport, London, 2006.}

For the later measures in Milan so vibrantly described we might see another public health angle:

In Milan and Venice stringent quarantine measures were initiated on the population. Infected people were isolated outside the cities, corpses and linens burned, and rooms and furniture were exposed to the sun for long periods of time. Once a person became infected, all occupants of his/her house, sick or well, were walled up and left to die. This was one of the worst possible ways of fighting plague. Anytime you group sick people with well, you increase the risk of infecting larger numbers of healthy individuals.
[…]
In most towns and cities throughout medieval Europe there existed local councils that were responsible for a wide array of duties, including health related issues. In Milan, during the 14th century there were a total of six officials that were responsible for environmental sanitation and street cleaning. Guilds also played a major role in medieval society, including carrying out health measures. Public health measures of the day were developed by lay people, who were not physicians nor affiliated with the church. Physicians were consulted on medical matters that involved legal issues and were employed to care for the indigent and those in prisons.
_{— Andrew Scott Warren: "Examination of Black Death and Public Health Implications for Today", UCHC Graduate School Masters Theses 2003–2010, 127, June 2001. (link)}

That these extreme experiments seem to have worked in Milan in 1348 despite absence of scientific evidence was of course no guarantee whatsoever that a neurotic 'more of these' would work anytime further down the line, despite continuity in leadership and actions:

Given its central position and high level of traffic, one would expect a catastrophe in 1348. In fact, Milan was but lightly brushed: Sienese chronicler Agnolo di Tura reported a mere three city families having been affected. The ducal government guarded every gate and screened every visitor, and when plague nonetheless appeared, the families were tightly sealed (shut in) in their homes. Similar prophylactic measures seem to have worked during the 1360 epidemic, but despite continued attention and experimentation, Milan suffered thereafter. Duke Giangaleazzo tried expelling all sick from subject Reggio in 1374, and extramural huts (mansiones) were provided to house victims, an early form of pest house. When plague approached the duchy in 1399, Milanese towns were cut off from contact with the capital; fairs and other gatherings were forbidden; victims and families were isolated in mansiones, and their houses fumigated. Later, hospitals were designated; victims were forcibly gathered and conveyed in carts. From 1424, a permanent ducal health commissioner oversaw this increasingly wide range of measures, including death registries with cause of death, from 1452. From 1468, Milanese authorities diagnosed each death and plotted geographic patterns, a century ahead of others. The Villa at Cusago became a plague hospice in 1447 and, in 1451, a plague hospital that served many of the epidemic’s 30,000 dead. During plague year 1468, authorities agreed to build a monumental lazaretto, San Gregorio, which was substantially completed two decades later, following the devastating epidemic of 1485. Following plagues in 1503 and 1523, it functioned fully in 1524. Truly dreadful but well chronicled plagues struck in 1576 to 1577 and 1629 to 1630, the latter brought by troops from the north. Archbishops Charles and Federigo Borromeo, uncle and nephew, actively led efforts to alleviate suffering. Charles built chapels and led processions, while Federigo did much the same and recorded his observations for posterity. Two centuries later, novelist Carlo Manzoni penned Italy’s “national novel,” in which the 1630 plague is a virtual character thwarting the protagonists. In 1629, San Gregorio hosted 10,000 plague victims and held another 15,000 at one time in 1630. From 1629 to 1636, Milan lost 60,000 of a population of 130,000.
_{— "Milan, Italy" in: Joseph P. Byrne (Ed): "Encyclopedia of the Black Death", 2012 ABC-CLIO: Santa Barabera, Denver, 2012, p236–237.}

Now that hopefully the reader is immunised partly against a short-circuited 'authoritarian extreme measures must have worked then', we have to emphasise most strongly, that lay people and experts back then really did not know what they were doing.

Among the prescriptions done we see one deemed as particularly much as effective as publicly mandated gauze cloth masking or breathing in bad smells:

The advice of the reverend father Dom Theophilus of Milan, of the order of St Benedict, against the plague; also a most wholesome medicine against all infirmities. Note it well.
Whenever anyone is struck down by the plague they should immediately provide themselves with a medicine like this. Let him first gather as much as he can of bitter loathing towards the sins committed by him, and the same quantity of true contrition of heart, and mix the two into an ointment with the water of tears. Then let him make a vomit of frank and honest confession, by which he shall be purged of the pestilential poison of sin, and the boil of his vices shall be totally liquified and melt away. Then the spirit, formerly weighed down by the plague of sin, will be left all light and full of blessed joy. Afterwards let him take the most delightful and precious medicine: the body of our lord and saviour Jesus Christ. And finally let him have himself anointed on the seat of his bodily senses with holy oil. And in a little while he will pass from transient life to the incorruptible country of eternal life, safe from plague and all other infirmities.
Compared with this all other remedies of doctors are futile and profit little against the plague, which God keeps for the chastis*m*nt of sin and which is without remedy save through him and his power.
— Rosemary Horrox (Ed & transl): "The Black Death", Manchester University Press: Manchester, New York, 1994.

And even today there are competing theories as to what was going on exactly in 1348–1351.

There are alternative hypothesis for explanations, like for example:

Seventh, critics point out that over time the lethality of the medieval pestilence dropped for no clear reason linked to modern understanding of bubonic plague. Unlike modern medicine, medieval medicine did nothing effective to prevent or cure cases of bubonic plague. A logical reply is that people became immune or resistant to bubonic plague by previous exposure in nonlethal doses: the body was able to develop the antibodies necessary to fight off the bacillus. Ell claims “survivors of plague infection have a potent immunity,” but Biraben, Carmichael, and Benedictow disagree, or at least they would qualify the claim by saying that immunity is short-lived: depending on the individual “from some months to several years.” Cohn flatly denies the possibility, stating that modern “human hosts have no natural or acquired immunity.” Ell, however, also states that typhus and at least one form of leprosy confer immunity to bubonic plague, as do various species of salmonella, a cause of food poisoning that, he notes, “may safely be assumed to have been ubiquitous” in medieval Europe. An earlier outbreak of typhus may have immunized many Milanese in 1348, for indeed the city of Milan was barely touched by the pestilence. Critics of “Black Death as bubonic plague,” however, dismiss any notion of large-scale immunization. They also traditionally disbelieved the idea that the bubonic plague bacillus mutated on a huge scale, so that its lethality and virulence would have dropped over time. The picture, however, is changing.

Completed research on and mapping of the Y. pestis genome sequence was reported in 2001 and showed that genetically it is very dynamic and fluid and subject to change. Scott and Duncan claim that only a slight mutation in Y. pestis can cause hypervirulence and potential epidemic by triggering the blocking mechanism in the flea’s proventriculus. They echo microbiologist Richard Lenski, who points out that high virulence kills off hosts, which can thus isolate the mutation. Virulence thus drops as the number of potential hosts drops. Less virulent forms are favored naturally, since they do not kill off their host population to the same degree. This might explain why recurrences of the pestilence tended to have lower mortality rates than the first outbreak. Scott and Duncan do not believe that any plague caused by Y. pestis appeared in England, but they do accept its presence on the continent.
[…]
Petrarch remained in northern Italy, and he decided to stay in Milan when the Plague hit again in 1361. When invited to flee to a country estate to wait out the carnage, Petrarch declined, claiming stoically that “to face [death] in fear is a base weakness.” His twenty-five-year-old illegitimate son, Giovanni, succumbed in July: “He died in Milan in the unexampled devastation wrought by the plague, which hitherto had left that city immune from such evils, but now has found it and invaded it.”

_{— Joseph P. Byrne: "The Black Death", Greenwood Guides to Historic Events of the Medieval World, Westport, London, 2004.}

So, whether they were very effective overall with certain 'measures' or not, whether this had any discernible effect on the spread of the plague – these are measurements of presumed effectiveness which are not easily accessible to us. Among range of plausible cause and effect relationships, a number of unexplained and unexplainable (yet?) factors have to be combined with the amount of sheer luck on display in various settings. It seems that the illusion of control is another element severely underemphasised in such an analysis.

What we can do more reliably is take a look at the historical record of what was done in general, as looked through the lens of 'still seen as perhaps moderately effective and not complete waste of time and resources':

The Italian system of plague defence had two main elements:

1. Communication. The northern Italian cities kept each other informed of the believed state of health of other locations in the region.
2. Defensive isolation. When contagious disease was uncovered anywhere by a particular Magistracy, a proclamation of ban (when the presence of communicable disease was positively ascertained) or suspension (precaution because there was legitimate suspicion of disease) was issued. Bans were long-term, suspensions short-term. Bans and suspensions were used to denote the interruption of regular trade and communication.

How effective these were over the course of time in Italy is then another question, perhaps illustrated by these pictures:

^{Plague outbreaks in Italy, 1347–1816. Annual time series of number of localities reporting plague. The generalized epidemics of 1348, 1383, 1457, 1478, 1522–28, 1577, 1630 and 1656 stand out from the annual background of 4–5 outbreaks which occurred in one place or another throughout the period. Source: data in Biraben (1975–76, Annexes III and IV, pp. 363–74, 394–400).}

^{Geographical distribution of number of recorded plague outbreaks in Italy, 1340–1820. Notes: (A) 1340–1450. (B) 1451–1550. (C) 1551–1650. (D) 1651–1820. Outbreaks declined in number and became more widely dispersed spatially over the period. Source: data in Biraben (1975–76, Annexe IV, pp. 394–400).}
_{— Andrew D. Cliff, Matthew R. Smallman-Raynor, Peta M. Stevens: "Controlling the Geographical Spread of Infectious Disease: Plague In Italy, 1347–1851", Acta med-hist Adriat 2009;7(1);197-236.}

As noted elsewhere, "barely touched" during the Black Death episode still means a 15% loss of population, despite the extreme measures with a less than perfectly explainable impact on possible modes of transmission:

Milan was the principal city on the Lombard plain. It controlled much of the overland Alpine trade with northern Europe and had close to 100,000 people in 1348. Like Genoa, Florence, Rome, and Venice, it was one of the major cities of Italy. Milan differed from the other centers, however, in the nature of its government. Its ruler was an absolute despot, a member of the Visconti family whose powers were more extensive than those of any contemporary ruler. When news of the Black Death reached Milan, the Viscontis and their advisers acted quickly. Municipal authorities walled up those houses in which plague victims were discovered, isolating in them the healthy as well as the sick. So popular did this become that many householders followed suit, in some cases killing members of their own families. Given the plague’s most common means of transmission, such measures should have had limited effect on mortality. Yet, Milan’s death rate was less than 15%, probably the lowest in Italy save a few Alpine villages. But Milan was exceptional. Generally, Italy, Europe’s commercial nexus, suffered as severely as it did because of its many points of entry for different strains of plague. Conservative estimates of mortality are about 33%, but many scholars believe that it reached 40% or even 50%. Taking into account the famines of the early fourteenth century, it is likely that the Italian population was reduced by 50%-60% from 1290 to 1360.
_{— Robert S. Gottfried: "The black death; natural and human disaster in Medieval Europe", Free Press: New York, London, 1983.}

To summarise this event regarding Milan and the supposedly effective measures taken under the auspices of Luchino Visconti:

The draconian Viscontis might have ruled the city, but it was not struck hard by the Black Death. For the Lord in power during the plague in the city was Luchino Visconti, a pragmatic and adaptable man, who used simple countermeasures such as quarantining the city, and blockading the houses of the sick, so they died without infecting anybody else. Without any cure or medicines, it was by far the cheapest and easiest way of dealing with the plague.
Even though the countermeasures were used, there seems to have been some deaths in the city. Armstrong claims that Milan suffered a 15% death toll which is not unreasonable to expect. But while a 15% death rate would have been devastating to a modern city, compared to contemporary cities they fared reasonably well. As a cause of the lack of death in the city, the Viscontis were not hampered severely by the plague. From all my research, it seems that during the Black Death, none of the Viscontis regarded plague. The Visconti seemingly cared more for their schemes and how they could gain more power. _{— Mads Ilebekk-Johansen: "To what extent was Milan affected by the Black Death from 1348-1350?", Essay, 2019}

From that we see that other areas were spared from the black death despite no comparative measures taken, whether we deem them possibly effective or not. And we see other areas that – unlike Venice, the most often cited counterexample to Milan – too some well thought out measures accounting for everything of contemporary latest fashions in medicine, based on mainly miasmatic theory, similar to Milan, but which suffered a great deal more than Milan. In contrast to Milan we have the best known example of Pistoria, which in seeing the plague coming enacted a much wider ranging regimen of even preventive public health measures, enacted in Spring of 1348: Pistoia, "Ordinances for Sanitation in a Time of Mortality".

These measurements were grounded in received wisdom and experience, easily comparable to Milan, or slightly later Venice or Florence, and yet they were quite ineffective in reducing the death toll during that crisis, which amounted to 25% of the population. _{( — G Geltner: "The Path to Pistoia: Urban Hygiene Before the Black Death", Past & Present, Volume 246, Issue 1, February 2020, Pages 3–33. doi)}

What remains in light of the above to describe Milanese leadership:

Intrigues, scheming games, political murders: This is how the ruling house of Milan spent those years, in which the plague depopulated Europe and wide regions became victims to the Black Death.
_{— Gesa Gottschalk: “Die Herrlichkeit”, Geo Epoche - 'Die Pest', No 139, 2015. Original Quote "Intrigen, Ränkespiele, politische Morde: So verbringt Mailands Herrscherhaus jene Jahre, in denen die Pest Europa entvölkert, weite Regionen dem Schwarzen Tod zum Opfer fallen."}

The map in question is the result of a distorting influence of the map made by Elisabeth Carpentier in here article: "Autour de la peste noire: Famines et épidémies dans l'histoire du XIVe siècle (Annales, pp. 1062-1092, 1962), analysed in "A Plague on Bohemia. Mapping the Black Death" (Excerpts: 'The power of a map.)

Although it is well worth the read, the small print on Wikipedia for other versions of this map gives links to multiple alternative versions:

Spreading of the en:Black Death in Europe between 1347 and 1351, apparently copied from a work called "Atlas zur Weltgeschichte" (perhaps dtv-Atlas? If so, Kinder and Hilgemann (eds.), dtv-Atlas zur Weltgeschichte was published in 35(!) editions between 1964 and 2002, and it would be crucial to know which edition we are basing this on). The details of the map are to be taken with a large grain of salt. It is roughly compatible with this Britannica map (copy, dated 1994), but the details vary significantly. Other significant differences exist with this map (an unreferenced scan of a professional publication), where e.g. the area of Poland/Silesia marked "unaffected" in this map is part of a much larger area marked as "Area for which there is insufficient information". Here is another map with a somewhat more conservative epistemology, referenced to "insecta-inspecta.com", where it was posted in 2000 (and therefore uninfluenced by the 2005 upload of this map). An academically published map for comparison: [1] cited to: D. Sherman and J. Salisbury, The West in the World: Volume I to 1715. McGraw-Hill, Boston, 3rd edition (2008), not necessarily better than the Britannica map, as it is a rough sketched used for the purposes of some argument about network theory, and not a medievalist publication dedicated to the history of the Black Death as such.

Since the initial map in question is as manipulative as the cherry picking quotations regarding an apparent success in Milan that disregards equal 'successes' in less affected areas and disregards its failures of the exact same measures when continued later, a perhaps better map would look like this:

_{— D. Cesana, O.J. Benedictow, R. Bianucci: "The origin and early spread of the Black Death in Italy: first evidence of plague victims from 14th-century Liguria (northern Italy)", Anthropological Science, Vol 125, No 1, 2017. doi}

Which is to say for the original map:

Though the map suggests the areas around Milan, Krakow, and the Spain-France border were unaffected, that's not actually true. [It] doesn't mean they objectively suffered a small number of deaths: According to Rutgers historian Robert Gottfried, the death rate in Milan was "only" 15 percent, and "only" around 25 percent in modern Poland. In other words, the rest of Europe was hit so badly that losing 15 percent of your population wasn't even enough to put a city on the map. (vox)

Summary:

The assumption that areas of Europe were principally 'spared' by the Black Death, when in reality they were not in the first wave and surely not in the subsequent waves, even if we construct an artificial separation between epidemics at a cutoff for 1351–1353.

Public health reactions in for example Pistoia, Florence, Venice and Milan were of comparable extent regarding modern understanding of quarantine and isolation, yet the outcome between 1347–1353 was different and the presumed successes of the Milanese approach, not so different from the others, was largely kept and expanded upon down the timeline. But such a success was not to be repeated. We see instead a flurry of frantic actionism, ever tighter controls in hope of doing and achieving 'something'.

Since we ourselves now do not even understand the true nature of the disease that ravaged the continent – a Y pestis bubonic plague origin is merely the currently still most popular hypothesis – it is also moot to ask the counterfactual: 'which measures really did or would have worked, were they adopted, enforced and kept'.