WHY: THE SOURCE OF THE CUSTOM
There is a widespread minhag (custom) to eat dairy foods on Shavuos (Remah OC 494:3). In a previous responsa we discussed how to balance this with the Mitzvah to eat meat on Yom Tov. Below are some of the reasons for this minhag (the why?), followed by the laws of eating meat after dairy (the how?).
The most commonly known explanation is that upon receiving the Torah on Shavuos, the Jewish nation became bound by the laws of Kosher. This meant that there was no time to prepare special shechitah (slaughtering) knives, remove blood and non-kosher fats from meat, and to kasher their utensils for hot meaty usage. This meant that they were unable to eat meat on the day when we received the Torah. Instead, on that first Shavuos our ancestors ate cold dairy foods, which we commemorate each year by doing likewise (Mishnah Berurah Ibid. sk.12). It's interesting to note that this reason only works according to the minority opinion that we received the Torah on a Friday (Pirkei DeRebi Eliezer 46:1). In the Talmud Bavli (Shabbos 86b) however, all agree, that we received the Torah on a Shabbos when they couldn't prepare such foods regardless! (This is noted in the Wagschal edition of the Mishna Berrurah Ibid.)
Eating dairy and then meat in one meal will necessitate replacing the bread on our table (which represents the altar). These breads serve as a reminder for the 'Korban Shtei Halecham', the Two Bread Sacrifice that was brought in the Beis Hamikdash (temple) on Shavuos (Remah Ibid).
Moshe Rabbeinu was taken out of the Nile on Shavuos and was thereafter brought to be nursed, and he refused to drink milk from non-Jewish women (quoted in Yalukut Yosef Ibid).
The gematria (numerical value) of the word 'חלב' - milk is 40, corresponding to the forty days that Moshe was on Har Sinai (Sefer Hamatamim p.30).
One of the names of Har Sinai is Gavnunim, similar to the word 'Gevinah' meaning cheese (Otzar Dinim Uminhagim p.394).
Some eat both honey and milk on Shavuos, as in Shir Hashirim (4:11) the Torah is compared to honey and milk (Chok Yaakov OC 494:9 from the Kol Bo s. 52).
It's brought from the Zohar that the seven weeks from Pesach to Shavuos, like the 'seven purification days', are connected to the Kabbalistic concepts of blood and milk which represents G-d's measure of judgment turning into mercy (see Magen Avrohom Ibid sk.5).
Rav Yitzchak Yosef (Yalkut Yosef Ibid) quotes another 16 reasons for this custom (see all 23 reasons quoted here)!
Overall, the custom of eating dairy foods on Shavuos remains cryptic and is not mentioned by many halachic sources. Regardless, our sages teach us that 'Minhag Yisrael Torah' - an accepted Jewish custom is Torah and, as with all holy customs, should be respected within the parameters of Halacha (Beer Hetev Ibid sk.8).
HOW: EATING DAIRY FOLLOWED BY MEAT
STEP ONE: CLEANING MOUTH
After eating dairy foods one must a) cleanse and b) rinse their mouth prior to consuming meat (as opposed to poultry). These two steps are called a) 'Kinuach' - cleansing the mouth and b) 'Hadochah' - washing the mouth. This is codified in the Shulchan Aruch (Yoreh Deah 89:2) as below:
"One must cleanse 'Kinuach' and rinse 'Hadochah' one's mouth; cleansing involves chewing bread, thereby cleansing the mouth very well. One may perform this with anything that he desires, except for flour, dates, and vegetables, since they adhere to the gums and do not cleanse well. And then one must rinse his mouth with water or wine. This is only for domesticated or wild aminal meats, but for poultry, there is no need for any cleaning or washing of hands."
STEP TWO: WASHING HANDS
In addition to the above, one must ascertain that their hands are clean. The Shulchan Aruch (Ibid) allows visually inspecting one's hands, however, in practice, one should not rely on this and instead, always wash one's hands after eating dairy before partaking of meat. This is because one cannot really tell if their hands are truly free of residue by merely looking at them (Shach Ibid sk.9 from the Rif and Iturei Zahav who states that this is the common custom).
If one used cutlery to eat the dairy foods (or drunk milk using a cup) and is certain that no dairy touched their hands, one need not wash their hands before eating meat (Be’er Hetev Ibid sk.5 from the Pri Chodosh, Aruch HaShulchan 89:8). One must be very careful with this exception, as in the course of eating, serving or cleaning up after a meal most often food somehow ends up on one’s hands.
DOES THE ORDER MATTER?
Although the Shulchan Aruch rules that one must first clean one's mouth and then wash one's hands, most opinions contend that the order does not matter (Shach Ibid sk.13, Be’er Hetev sk.7). As such, in practice one need not be concerned about the order.
WHAT ABOUT POULTY?
The above steps pertain only to one who ate a dairy meal and then wishes to eat ”meat” in the true sense of the word, such as beef, veal, or venison. Poultry requires no cleansing and rinsing of the mouth or washing of hands when eaten after dairy foods (as quoted in the Shulchan Aruch above).
IS THERE A WAITING PERIOD?
The Gemara (Chullin 105a) quotes Rav Chisda, who states that one need not wait at all after eating cheese before consuming meat. Indeed, the Shulchan Aruch (Ibid) rules that one may eat meat immediately after cleaning and washing, with the other major Poskim also not recording any waiting period (Rif, Rambam, and Tur).
However, as alluded to in a previous responsa the Zohar in (Parshas Mishpatim 155a) indicates that one should not eat dairy and meat in one meal and must recite the Bracha Acharonah (Grace after Meals) after a dairy meal and then wait before being permitted to consume meat. Although this isn't required Halachically, many conduct themselves as such and wait half an hour or an hour in light of this.
However, after eating hard cheeses which have aged over six months there is a minimum waiting period of six hours (i.e. as is your custom to wait between meat and milk) prior to eating meat (Remah YD 89:2 as explained by Shach sk.16). Others note that regardless of the cheese’s age, one should also wait after eating pungent, strong-tasting cheeses (Taz Ibid. sk.4).
For questions regarding specific cheeses, see the Orthodox Union's useful list of aged cheeses www.oukosher.org/blog/consumer-kosher/aged-cheese-list.