Preheat the oven to 375 degrees. 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 ½-inch-thick slices, and place in a colander. Sprinkle with the 1 tablespoon salt, and let drain for 1 hour. Rinse under cool running water, drain thoroughly, and pat dry. Whisk the eggs in a wide, shallow bowl, and season with salt and pepper. Spread the flour out evenly in a wide, shallow bowl. Dredge the eggplant slices in flour, shaking off the excess. Dip the floured eggplant into the eggs, and set in a colander to drain until ready to fry.
Meanwhile, pour ½ cup each of the vegetable and olive oils into a large 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 browned on both sides, about 4 to 6 minutes total. Remove the eggplant to a baking pan lined with paper towels and repeat with the remaining eggplant slices. Add oil to the pan as necessary during frying to keep the level the same.
Bring the tomato sauce to a simmer in a saucepan. Add shredded basil. Ladle enough sauce into a 9-by-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. Ladle about 1 cup of the sauce over the the eggplant to coat the top evenly. Sprinkle a third of the grated cheese over the sauce, and top with a third of the mozzarella or Fontina, spreading it evenly over the eggplant. Repeat the layering as described above two more times, ending with a top layer of mozzarella or Fontina cheese.
Drizzle sauce around the border of the baking dish, and sprinkle the top with the remaining grated cheese. Drizzle any remaining sauce on top, cover with foil, and bake until it is bubbling and golden in places, about 30 to 35 minutes. Uncover, and continue baking until golden brown all over, about 15 minutes. Let it rest 10 to 20 minutes, then cut it into squares and serve.