Position racks in the upper and lower thirds of the oven and preheat to 450 degrees . Coat two baking sheets with cooking spray.
On a large plate, combine the flour and pepper. In a large bowl, whisk the eggs and egg white. On another large plate, spread out the breadcrumbs. Working with one eggplant slice at a time, coat with the flour, then the egg mixture, then the breadcrumbs, shaking off the excess after each step. Place on prepared baking sheets. Repeat with the remaining eggplant.
Bake 10 minutes, then flip the eggplant slices and switch the positions of the pans in the oven. Bake until the eggplant is golden-brown, about 10 minutes more.
Coat an 8-inch-square baking dish with cooking spray, then spread with 1/2 cup sauce. Arrange one-third of the eggplant slices in a single layer, cutting slices as needed to fit the pan. Top with 1/2 cup sauce, 1/2 cup mozzarella and 2 tbsp. Parmesan. Repeat the layering twice more with the remaining ingredients, smoothing any remaining sauce on top.
Reduce the oven temperature to 350 degrees . Bake until the cheese melts and the casserole is browned and heated through, 25 to 30 minutes. Let rest 5 minutes before serving.