A piece of tofu is rolled in katakuriko (Potato Flour — sometimes erroneously translated as potato starch), then deep fried. The tofu piece gets a crispy shell on the outside and gets very hot on the inside.
It is best made with Silken Tofu. You cook the tofu very briefly in hot water just to remove the raw taste (do this by putting it in a pan of water that you bring to the boil, then remove from the heat, then plunge the tofu into cold water.) Then sit the tofu on paper towel or a tea towel to drain it for about half an hour, then dredge it in whatever you are using for flour, and fry until golden. Instead of the potato flour, cornstarch or kudzu starch can be used.
It is served lightly garnished with something such as chopped ginger and green onion, or grated daikon radish, green onion and momiji oroshi (chile powder.) Sometimes it is garnished with bonito tuna flakes (katsuobushi.) Which garnish is used depends which sauce it is being served with.
The sauce accompanying it will vary. It can be a ponzu sauce thickened with starch, or a sauce based on light or dark soy sauce (usukuchi or koikuchi shoyu), or dashi, mixed with other ingredients such as mirin, sake, soy sauce and / or sugar. Sometimes the sauce is served on the side; sometimes the sauce is poured over it.
“A-ge” means deep-fried; “dashi” is the broth.