Peel eggplant and slice into rounds.  Batter each slice by dipping into
 beaten egg, then dredging in flour.  Fry the battered slices in vegetable
 oil, flipping so that both sides turn a nice golden brown.
 Layer a casserole dish with the eggplant slices, grated parmesan and/or
 mozarella cheese, and marinara sauce (see below).  Dust top with grated
 parmesan, then bake in 350F oven until the top turns slightly brown and
 bubbly.  Allow to cool some before serving.
 Marinara sauce:  Saute several cloves minced garlic and a chopped small
 onion in several tablespoons olive oil until translucent.  Add 3-4 cups
 peeled crushed tomatoes (I usually use canned), 1/4 cup red wine, and a
 bay leaf.  Turn down the heat and let simmer.  After 10 minutes or so,
 add salt and black pepper to taste, sweet basil, oregano, and crushed
 red pepper flakes (optional).  Continue to simmer, the longer the better,
 adding red wine or water if the sauce becomes too thick.
 A simplified version, perhaps closer to an actual “au gratin”, would be
 to simply grill or saute the eggplant slices without batter, layer with
 cheese, and bake as above.