1. Milchigs. Some say one should eat cheese and milchig dishes on Chanukoh since the miracle was brought about by the milk Yehudis gave to the enemy (רמ"א תר"ע ס"ב ).
2. Donuts. There is an old minhag to eat donuts on Chanukoh since they are fried in oil, which recalls the miracle of the olive oil jug (רבינו מיימון בן יוסף אבי הרמב"ם נדפס בקובץ שריד ופליט ירושלים תש"ה ). Eating milchige donuts fulfills both minhagim simultaneously.
3. Rav Shlomo Zalman Auerbach offered another reason: when the Chashmonaim were purifying the Beis Hamikdosh, they did not know how to purify the stones of the mizbeiach that the Greeks defiled, so they hid them away (ע"ז נ"ב: ). Therefore, we eat food that requires Birkas Mei'ein Sholosh, which, unlike Birkas Hamozon, contains the words, "ועל מזבחך ." For this reason, we specifically eat donuts outside of a bread meal in order to say Al Hamichyoh (הליכות שלמה ארחות הלכה ).
