When I bake and fry things like these slices of eggplants, I make a little assembly line that leads from the flour to the eggs, on to the breadcrumbs and right into the pan of hot oil. Placing three rectangular cake pans side by side next to the stove works nicely—there is very little cleanup afterwards—but any container wide enough to hold several slices of eggplant at a time will work just as well.
This dish can be made with roasted eggplant slices instead of breaded and fried eggplant. Although it will be good, it will not be as tasty nor will it have the texture of the fried eggplant. The roasted version is very simple: drain and rinse the eggplant as described above, but instead of coating the eggplant slices, toss them with a few tablespoons of olive oil. Brush a baking sheet with olive oil, and set the eggplant slices side by side on the baking sheet. Bake them in a 450° F preheated oven for 20 minutes till they are golden brown. Let them cool and proceed to layer and bake the ingredients as above.
Trim the stems and ends from the eggplants. Remove strips of peel about 1-inch wide from the eggplants, leaving about half the peel intact. Cut the eggplant lengthwise into 1/2-inch thick slices and place them in a colander. Sprinkle with the coarse salt and let drain for 1 hour. Rinse the eggplant under cool running water, drain throroughly and pat dry.
Whisk the eggs and 1 teaspoon salt together in a 13 x 9 inch baking pan or wide, shallow bowl. Spread the flour and breadcrumbs in an even layer in two separate wide, shallow bowls or over sheets of wax paper. Dredge the eggplant slices in flour, shaking off the excess. Dip the floured eggplant into the egg mixture, turning well to coat both sides evenly. Let excess egg drip back into the pan, then lay the eggplant in the pan of breadcrumbs. Turn to coat both sides well with breadcrumbs, pressing with your hands until the breadcrumbs adhere well to the eggplant.
Pour 1/2 cup each of the olive oil and vegetable oils into a medium skillet. Heat over medium-high heat until a corner of the eggplant slices gives off a lively sizzle when dipped into the oil. Add as many of the eggplants slices as fit without touching and cook, turning once, until well browned on both sides, about 6 minutes. Remove the eggplant to a baking pan lined with paper towels and repeat with the remaining eggplant slices.
Adjust the heat as the eggplant cooks to prevent the bits of coating that fall off the eggplant slices from burning. Add the oil to the pan as necessary during cooking to keep level more or less the same.
Preheat the oven to 375F. Heat the tomato sauce too simmering, if necessary, in a small saucepan over medium heat. Ladle enough sauce into a 9x13inch bakign dish to cover the bottom. Sprinkle with an even layer of grated cheese and top with a layer of fried eggplant, pressing it down gently.
Tear a few leaves of basil over the eggplant and ladle about 3/4 cups of the sauce to coat the top evenly. Sprinkle an even layer of grated cheese over the sauce and top with a layer of Mozzarella or Fontina, using about one-third of the cheese.
Repeat the layering as described above two more times, ending with a top layer of sliced cheese that leaves a border of about 1 inch around the edges of the baking dish. Drizzle sauce around the border of the baking dish and sprinkle the top layer with the remaining grated cheese. Finish with a few decorative streaks or rounds of tomato sauce. Cover the baking dish loosely with aluminum foil and poke several holes in the foil with the tip of a knife. Bake 30 minutes.
Uncover, and continue baking until the top layer of cheese is golden in spots about 15 minutes. Let rest 10 to 20 minutes, then cut into squares and serve.