Preheat oven to 400°F. Lightly coat a jelly roll pan and one 8-by-11 1/2-inch baking dish with olive oil (we used olive oil spray).
Cut eggplant crosswise into 1/4-inch-thick slices. Whisk egg white and water in a shallow dish until frothy. Combine breadcrumbs, 1/4 cup of the Parmesan, and pepper in another shallow dish. Dip the eggplant slices into the egg-white mixture, then coat with the breadcrumb mixture. Arrange the eggplant slices in a single layer on the prepared jelly roll pan. Bake for 15 minutes, turn the eggplant slices over, and bake until crisp and golden, about 15 minutes longer. Stir fresh basil into tomato sauce. Spread about 1/2 cup of the sauce in the bottom of the prepared baking dish. Arrange half of the eggplant slices over the sauce, overlapping slightly. Spoon 1 cup of the remaining sauce over the eggplant and sprinkle with half of the mozzarella cheese. Add a layer of the remaining eggplant slices and top with the remaining sauce, mozzarella and Parmesan or Asiago cheese.
Bake, uncovered, until the sauce bubbles and the top is golden, 15 to 20 minutes. 4 servings.
*We made a simple tomato sauce with 1 diced onion, sautéed in 1 TBS olive oil; 3 large tomatoes, peeled and diced; 1 clove garlic, pressed; 1 8-oz can of tomato sauce (with no salt added); red pepper flakes and Italian seasoning, to taste. Sauté the diced onion in olive oil until soft, add diced tomatoes, cover and simmer gently for ~20 minutes. Add tomato sauce, thin to desired consistency with water, and season to taste.