1. Make the tomato sauce: In a large skillet, heat the olive oil over medium heat for 1 minute. Add the garlic and cook just until it starts to sizzle, about 1 minute. Do not let the garlic brown. Add the tomatoes with their juice and use a fork to break them up into pieces ½ to ¾ inch in size. Adjust the heat so the tomato juices are bubbling gently and cook, uncovered, for 15 minutes, stirring occasionally. Add the basil and cook another 5 minutes. The sauce will reduce slightly.

2. Remove the pan from the heat, season with salt, and set aside. The sauce can be made ahead, cooled, sealed in a container, and stored in the refrigerator for up to 3 days.

3. Make the filling: Trim off the stems and base of the eggplants and cut them into ½-inch crosswise slices. Layer the slices in a colander, sprinkling each layer generously with salt. Top with a plate, then a weight (a large unopened can works well), and let sit for at least 20 minutes. You will see moisture form on the eggplant slices. Rinse off the salt and use paper towels to pat the slices dry.

4. Position a rack in the middle of the oven. Preheat the oven to 350°F. Have ready a 9-by-13-by-2-inch rectangular or oval baking dish.

5. Lay a large piece of waxed paper on the counter. Line a baking sheet with several layers of paper towels and set aside. Put the flour on a large plate and season it with a sprinkling of salt and generous grindings of black pepper, stirring to distribute the salt and pepper. Dip both sides of each eggplant slice in the seasoned flour, shaking the excess flour gently back onto the plate, and arrange them in one layer on the waxed paper. In a large skillet, combine the 2 tablespoons of olive oil with enough corn or canola oil to reach a depth of ¼ inch. Heat the oil over medium heat for about 3 minutes, until a slice of eggplant sizzles and bubbles gently around the edges when added. Fry a single layer of eggplant slices, turning once, until golden, about 2 minutes on each side. Use a fork or tongs to turn and lift the slices. You will probably need to fry them in 2 or 3 batches, depending on the size of the pan. Drain the slices in a single layer on the paper towel-lined baking sheet.

6. Spread ½ cup of the tomato sauce over the bottom of the baking dish. Cover the bottom with half of the eggplant slices. Top with half of the mozzarella, 1 cup of the tomato sauce, and ¼ cup of the Parmesan cheese. Add another layer of the remaining eggplant slices, mozzarella, and sauce. Sprinkle the remaining ½ cup of Parmesan cheese over the top. Set aside.

7. Prepare the topping: In a medium bowl, stir the bread crumbs and Parmesan cheese together. Stir in a sprinkling of salt and a few grindings of pepper. Stir in the water. Stir in the olive oil until the crumbs are evenly moistened and cling together. Spoon the crumbs evenly over the filling and pat them gently to smooth the top. The crumbs should completely cover the filling.

8. Bake until the topping is lightly browned and crisp and the filling is bubbling gently, about 30 minutes. Use a large, sharp knife to cut through the layers to the bottom of the dish and cut the filling into squares. Use a wide spatula or large spoon to lift the pieces onto serving plates.