Sprinkle 1 teaspoon of salt evenly over the eggplant slices on a large baking sheet, cutting board, or directly in a bowl. Layer several sheets of paper towels on a large cutting board or baking sheet. Arrange about a third of the eggplant slices in a single layer, and top with more paper towels. Continue layering the eggplant slices and paper towels, ending with a final layer of paper towels. Let the eggplant sit for 1½ to 2 hours. Press on the paper towels to absorb excess moisture before proceeding.

Prepare the béchamel sauce: Melt the butter in a small saucepan over medium heat. Add the flour, whisking constantly for 2 minutes. Gradually whisk in the milk until smooth. Bring to a boil, then lower the heat and simmer, whisking frequently, for 2 minutes, until thickened. Stir in the Parmesan cheese, salt, and pepper. Set aside.

On a large plate, combine the flour with ¾ teaspoon salt. In a separate bowl, whisk the eggs with 2 tablespoons of cold water. On another plate, mix the breadcrumbs with ¾ teaspoon salt.

Dredge each eggplant slice in the flour, shaking off the excess. Dip in the egg mixture, allowing any excess to drip off, then coat with breadcrumbs, pressing to adhere. Arrange the breaded eggplant slices on a baking sheet.

Heat about ¼ inch of oil in a large (12-inch) skillet over medium heat until hot (it should sizzle when an eggplant slice is added). Fry the eggplant in batches, turning once, until golden brown, about 3 minutes per side. Transfer to a paper towel-lined baking sheet to drain. Continue with the remaining eggplant.

Preheat the oven to 425°F and position a rack in the center. Spread ¾ cup of marinara sauce on the bottom of a 9x13-inch baking dish. Arrange a third of the eggplant slices over the sauce, overlapping them as needed. Top with another ¾ cup of marinara sauce. Sprinkle with 1 cup of mozzarella cheese and spoon about one-third of the béchamel sauce over the cheese. Repeat two more layers, leaving the edges of the eggplant visible on the top layer for crispiness. Sprinkle with Parmesan cheese. Bake for 30 to 35 minutes until bubbling and golden brown. Let cool, loosely covered with foil, for about 20 minutes. Garnish with basil if desired, slice, and serve.

Note: Nutritional information assumes that about ½ teaspoon of salt used on the eggplant is absorbed and that approximately ¼ cup of oil is absorbed during frying.

Make-Ahead/Freezer-Friendly Instructions: The assembled dish can be refrigerated up to 2 days before baking or frozen, either before or after baking, for up to 3 months. If frozen, defrost in the refrigerator overnight before reheating or baking.