Make sauce: Saute onion with olive oil until golden in a medium saucepan. Add tomatoes and ½ teaspoon salt. Cook at a brisk simmer, breaking up tomatoes with a wooden spoon, until onion is cooked and tomatoes soften, about 10 minutes. Cool to warm. Using a blender or food processor, process to a chunky sauce.
Preheat oven to 450°F. Peel eggplant or not, depending on preference. Cut in ½-inch slices. Place flour in a shallow bowl and, in a second bowl, beat eggs with 1 teaspoon salt. Coat slices with flour on both sides, dusting off excess. Dip in egg, allowing excess to drop off.
Fill large deep skillet with at least ½ inch vegetable oil. Heat over medium-high heat until shimmering. Fry eggplant until golden brown on both sides. Drain on a platter lined with paper towels and blot with more paper towels.
Ladle sauce over bottom of a 13x9x2-inch baking pan (see Note). Arrange a layer of eggplant on top. Sprinkle with half the mozzarella, Parmigiano Reggiano and basil. Repeat layers, finishing with sauce and cheeses.
Bake uncovered for 25 to 30 minutes until bubbling and browned on top. Cool at least 15 minutes before cutting.
Recipe Notes
Note: In a smaller pan, you may need to repeat the sequence of layers three times rather than twice.