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 into1/2-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 thoroughly and pat dry.

Whisk the eggs and 1 teaspoon fine salt together in a 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.

Pour ½ cup each of the olive and vegetable oils into a medium skillet. Heat over medium-high heat until a corner of one of the eggplant slices gives off a lively sizzle when dipped into the oil. Add as many of the eggplant 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 towel 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 oil to the pan as necessary during cooking to keep the level the same.

Preheat the oven to 375 degrees F. Heat the tomato sauce to simmering, if cold or frozen, in a small saucepan over medium heat. Ladle enough sauce into a 9 x 13-inch baking 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 ¾ cup 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 cheese that leaves a border of about one 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, until the cheese is bubbling and golden in places.

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.