Preheat oven to 400°. Line two baking sheets with parchment paper. Oil a 9-by-12-inch baking dish with 2 tablespoons olive oil.
Slice eggplants into 1/4-inch-thick slices. You need to get 24 nice round slices (keep the thinner end and the tip to use in a stir fry or something else).
Whisk eggs and milk in a shallow dish until frothy. Combine breadcrumbs, 1/4 cup of the Parmesan, salt and pepper in another shallow dish. Dip the eggplant slices into the egg mixture, then coat with the breadcrumb mixture.
Arrange the eggplant slices in a single layer on the prepared baking sheets. Bake for 15 minutes, turn the eggplant slices over, and bake until crisp and golden, about 15 minutes longer.
Stir basil into tomato sauce. Spread about 1/2 cup of the sauce at the bottom of the prepared baking dish.
Arrange half of the eggplant slices over the sauce. Spoon 1 cup of tomato sauce over the eggplant and sprinkle with half of the mozzarella cheese.
Add a layer of the remaining eggplant slices and top with remaining sauce, mozzarella and Parmesan.
Bake, uncovered, until the sauce bubbles and the top is golden, 15 to 20 minutes. Let the dish rest for 5 minutes before eating as the tomato sauce will be very hot...