Since Shavuot is a time when Hilchot Basar Bechalav is commonly discussed, we will discuss the Ashkenazic custom of avoiding kashering from milk to meat and vice versa. We will focus on the sources and parameters of this practice.
This Minhag makes its first appearance in the Magen Avraham's comments to Shulchan Aruch Orach Chaim 509 (no. 11). This authority writes (note that he does not write "I believe this should be done" but rather he is recording %3&-. 1&%#*0 - common practice)
"Common practice is to forbid Kashering (-%#3*-) milk utensils to meat utensils and vice versa.... A good explanation of this practice is that if this would be permitted, people would own only one utensil and would kasher this utensil from meat to milk and vice versa which is forbidden because it would cause confusion."
The concern that people own only one utensil may seem to be far-fetched to us, but was a highly relevant concern for Eastern European Jews who by and large lived in dire poverty.
It should be noted that Rav Moshe Feinstein (Teshuvot Igrot Moshe Y.D. I:44) writes that this custom applies today even though we do not live in poverty and it is affordable for almost all to own two sets of dishes. The Magen Avraham cites the following Talmudic passage as a source for this practice (Chullin 8b)
"A butcher should own three knives: one for Shechting (Rashi: This knife should not be used for non Shechita purposes as it may cause nicks (5#*/&;) to develop on the knife), one for cutting meat, and one for cutting forbidden fats, (-"*.. Why, queries the Gemara, is one knife not sufficient for meat and (-"*., just clean the knife after use with (-"*.? The Gemara answers, because we are concerned that he will cut (-"*. and subsequently cut meat without cleaning the knife, we don't allow this. But, asks the Gemara, he still might make a mistake even with two knives! Answers the Gemara since two different knives are required, this serves as a reminder not to mistake one knife for another. Similarly, Jewish homes should have two sets of as an incentive cutlery to avoid confusion and mixups.
Not all authorities accept the view of the Magen Avraham. The Aruch Hashulchan (89:17) writes that it is a (&/9! *;*9! (excessive stringency) and that we lack
the authority to institute new prohibitions (see Rosh Shabbat 2:15 who states that new legislation cannot be enacted subsequent to the conclusion of the Talmudic era).
Indeed it is clear that the Shach (Yoreh Deah 121:8) permits Kashering meat utensils for use as milk utensils and vise versa. Moreover, Rav Ovadiah Yosef(Teshuvot Yabia Omer 3:Y.D.4) notes that the great Sephardic authority the Chida writes that Sephardim do not accept the practice. Nevertheless the common practice is to abide by this custom (see Mishna Berura 451:30). However, since this is only a minhag, the Poskim present many exceptions to this rule.
Exceptions to the Rule - Rendering the Utensil Non-Kosher
The Pri Megadim (Aishel Avraham to O.C. 509:17) notes an interesting Minhag which developed to circumvent this custom. The practice was to render the utensil Treif, thus one would be transforming a non-Kosher utensil to be a Kosher one and not changing a milk utensil to a meat one.
The Darkei Teshuva (121:59) cites authorities who criticize this ruling as a violation of the !*0 /)"-*0 !*2&9 -,;(*-% rule (that forbids intentionally nullifying forbidden foods). This, however, appears to be a less than convincing criticism as the practice involves circumventing the practice of not Kashering from meat to milk and not nullifying prohibited food (see Shulchan Aruch Y.D. 95:4). Moreover, engaging in such an unusual practice reinforce the idea that we should not routinely Kasher back and forth from milk to meat. The Yad Yehuda (97:36, a major commentary to Yoreh Deah) notes that the common practice is to follow the Pri Megadim. Rav Menachem Genack informed me that the great Rav Eliezer Silver (one of the great rabbinic leaders in this country in the mid-twentieth century) told him that this indeed was the accepted practice.
Kashering to Pareve- Libun
Poskim have limited the application of our minhag. For example, the Yad Yehuda and the Teshuvot Meharsham (2:241) permit Kashering either milk or meat utensils for pareve use. For example, if a husband used a Pareve knife for meat, the utensil may be Kashered with the intention of restoring its original status. The Darkei Teshuva (an early twentieth century compendium of the responsa literature on Yoreh Deah) cites no dissenting opinions who forbid Kashering from meat or milk to Pareve.
The Darkei Teshuva cites opinions who assert that this Minhag applies only to Kashering by boiling water (%#3-%) and not to Kashering through fire (-*"&0). This is presumably permissible due to the fact that the -*"&0 process is so difficult that one would not be in the habit of performing -*"&0 every time one wished to Kasher from milk to meat. Thus, one may Kasher an oven designated for milk use to be used for meat or vice versa, as Kashering the oven involves the -*"&0 process and not the %#3-% process.
Pesach Use - The Chatam Sofer
The Chatam Sofer writes (Teshuvot Y.D. 110) that when one Kashers for Pesach, one may change from milk to meat (this arises often as sometimes people forget which of their Pesach utensils are meat or milk). The reasoning is that since Kashering for Pesach is a once a year event it will not lead to forming a habit of regularly Kashering back and forth from milk to meat. The Mishna Berura (451:19) rules in accordance with this ruling of the Chatam Sofer.
Twelve Month Wait
A minority opinion believes that after twelve months, whatever is absorbed in utensils is considered like 359! $3-/!, dust of the earth. Thus, whatever food particles were absorbed in the utensil have lost all Halachic significance. According to this view, one need not Kasher a non-Kosher utensil that has not been used for twelve months (see Chochmat Adam 55:4 citing Teshuvot Chacham Zvi). Although this view is not the accepted view, Poskim use it as a lenient consideration (see Teshuvot Igrot Moshe Y.D. 2:46). Accordingly, it is not surprising to find that the Maharsham permits utensils that haven't been used for twelve months to be Kashered back and forth from milk to meat. His reasoning could be that the Minhag was not enacted for unusual circumstances, akin to what the Gemara often states /*-;! $-! :,*(! -! #'9& "*% 9"10 (the rabbis did not exact legislation for unusual circumstances).
Utensils that One Purchases or Receives as a Gift
The Darkei Teshuva quotes yet another exception to this Minhag. That is when one purchases or receives utensils as a gift he may Kasher these utensils with the intention of changing the dishes from milk to meat or vice versa. Again, since the acquiring of utensils is a unique event the concern that he will get into the habit of routinely Kashering from milk to meat and vice versa, is not relevant.
Kashering After an Accidental Spill
The Darkei Teshuva cites an anonymous halachic authority (possibly Rav Gershon Adler) who ruled that in case of accidental spillage, one may restore the utensil to its original status. His reason is that the Magen Avraham's concern is relevant only to intentional change of utensils from milk to meat. However, his concern is not relevant to a situation where one accidentally made a milk dish "meaty" through contact with a hot meaty food substance. Moreover, since the dish has absorbed both milk and meat taste particles the utensil may be viewed as "not Kosher" and thus we may Kasher it to restore the utensil to its original state. See the full discussion of this matter in the Darkei Teshuva, as well as Teshuvot Igrot Moshe Y. D. I:44. Of course, it goes without saying that one must consult his halachic advisor should any of these issues arise.