It is preferable to use oil for the lighting of the Hanukah candles, since the miracle which we celebrate involved the oil of the Menorah in the Temple. Specifically, it is preferable to use olive oil – the oil that was used for the kindling of the Menorah.
For the same reason, there is a widespread custom to eat fried foods on Hanukah, to commemorate the miracle of the oil. It is common to eat fried sweetened donuts, generally with jelly inside to enhance the sweet taste. The Beracha over jelly donuts is "Mezonot," regardless of how much one eats. When it comes to other "Mezonot" foods, one who eats a large quantity would recite "Ha’mosi" before eating and Birkat Ha’mazon after eating. Over a fried "Mezonot" food, however, one always recites "Mezonot" before eating and "Al Ha’mihya" after eating, even if he eats 3-4 donuts in a single sitting. If one eats a fried donut as dessert at a meal which included bread, then he does not recite a Beracha, as the donut is covered by the Beracha of "Ha’mosi" recited at the beginning of the meal.
Rav Shlomo Zalman Auerbach (Israel, 1910-1995) was asked whether one may warm a jelly donut on Shabbat by placing it on a hotplate or blech (metal sheet that covers the stove). It is permissible on Shabbat to reheat dry, fully cooked foods on these surfaces, but Halacha forbids reheating liquids on Shabbat in any fashion. It would thus seem, at first glance, that warming a jelly donut would be forbidden on Shabbat because the liquid jelly is reheated. Rav Shlomo Zalman, however, ruled that one may place a jelly donut on a hotplate or blech, because the jelly is concealed inside the donut, and in any event it constitutes the minority of the food, which is mostly solid. This is also the ruling of Hacham Ovadia Yosef.
It is customary to fry and eat potato pancakes, or "latkes," during Hanukah. What Beracha does one recite on latkes? Over potatoes, of course, one recites "Ha’adama," but latkes are made from grated or pureed potatoes, and the question arises as to whether this process affects the Beracha.
Sephardic custom is to recite "Ha’adama" even over mashed potatoes, and also over edg’ge, which resembles latkes. According to this custom, the process of crushing does not affect the Beracha, and thus one recites "Ha’adama" over potato latkes. Of course, if one eats latkes in a sandwich, he simply recites "Ha’mosi" over the bread and this Beracha covers the latkes.
Many people eat applesauce with the latkes to add a sweet flavor. Once again, according to Sephardic practice the Beracha remains unchanged despite the crushing process entailed, and thus one recites "Ha’etz" over applesauce just as one would recite over an apple.
Summary: It is customary to eat fried foods during Hanukah, specifically fried jelly donuts and fried potato pancakes ("latkes"). One recites "Mezonot" and Al Ha’mihya over jelly donuts even if he eats a full meal’s worth. It is permissible to warm jelly donuts on a hotplate or blech on Shabbat. According to Sephardic custom, one recites the Beracha of "Ha’adama" over potato pancakes.
