Preheat oven to 400°F. Coat two baking sheets and an 8-by-11 1/2-inch baking dish with nonstick cooking spray.
Cut eggplants crosswise into 1/4-inch-thick slices. Whisk egg whites and water 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-white mixture, then coat with the breadcrumb mixture. (Discard any leftover breadcrumbs and egg white.) 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 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. Bake, uncovered, until the sauce bubbles and the top is golden, 15 to 20 minutes.