Peel and cut a large eggplant into slices, dip in egg wash (1 egg beaten with 1 tablespoon water) then bread crumbs and fry until just golden brown. Place half of the fried eggplant slices in a baking dish and top with a slice of fresh, peeled tomato. Place remaining fried eggplant slices on top of those in baking dish and top with another slice of tomato. Top each eggplant and tomato "sandwich" with a slice of American cheese and place in a 350 degree preheated oven until cheese melts.