There is a well-documented custom to eat fried doughnuts on Chanukah, as part of our commemoration of the miracle that occurred with the jug of oil. This custom is recorded in many Halachic works, and the work "Kovetz Sarid U'palit" refers in this context to a comment by Rabbi Maimon, father of the Rambam, warning that we should not treat lightly any custom that has been accepted by the Jewish people.
Chacham Ovadia Yosef, in his work Chazon Ovadia (Laws of Chanukah, p. 18), writes that if one eats a fried doughnut with his meal, he should preferably eat it only after Birkat Ha'mazon, given the controversy that exists as to whether it requires a Beracha during a meal. If one does eat a doughnut during the meal, rather than waiting until after Birkat Ha'mazon, he does not recite a Beracha over the doughnut.
It is common for the fried doughnuts eaten on Chanukah to have a concentrated area of liquid jelly in the center. The presence of liquid jelly gives rise to the question of whether or not one may reheat a jelly doughnut on Shabbat, by placing it on a "blech" or hotplate. Halacha forbids reheating a cooked liquid on Shabbat; seemingly, then, it would appear that should not reheat a fried doughnut that has n area of liquid jelly in the center.
Rabbi Shlomo Zalman Auerbach (Jerusalem, 1910-1995) held that one may, in fact, reheat a jelly doughnut on Shabbat. Since the jelly is contained inside the doughnut, it is deemed part of the doughnut, which is solid. Hence, since one is permitted to reheat solid food items on Shabbat, one may reheat a jelly doughnut, as well. This is particularly so in light of Chacham Ovadia Yosef's ruling in his work Yabia Omer (vol. 7, Orach Chayim section, 42) that a food that is mostly solid may be reheated on Shabbat even if it contains some liquid. Certainly, then, one may reheat a jelly doughnut on Shabbat.
Rabbi Shlomo Zalman was also asked as to whether yeshiva students observing a period of Aveilut (mourning), Heaven forbid, may participate in the yeshiva's Chanukah festivities. He held that if no music is played at the celebration, it is permissible for mourners to participate.
Summary: There is a time-honored custom to eat fried doughnuts on Chanukah, and it is proper to observe this tradition. Preferably, doughnuts should be eaten only after Birkat Ha'mazon; if one eats a doughnut during a meal before Birkat Ha'mazon, he does not recite the Beracha of "Mezonot" over the doughnut. One may reheat a fried jelly doughnut on Shabbat on a "blech" or hotplate. Mourners may participate in Chanukah parties that do not feature musical accompaniment.
