Step 1 Salt the eggplant: In a large bowl, on a baking sheet, or directly on a cutting board, sprinkle 1 teaspoon of salt evenly all over the eggplant slices. Lay out a few layers of paper towels on a large cutting board or baking sheet. Arrange about one-third of the eggplant slices in a single layer and cover with more paper towels. Continue stacking the eggplant and paper towels, finishing with a final layer of paper towels. Let sit for 1½ to 2 hours. Press on the paper towels to absorb any excess liquid before proceeding.
Step 2 Make the béchamel sauce: Melt the butter in a small saucepan over medium heat. Add the flour and cook, whisking constantly, for 2 minutes. Add the milk and whisk until evenly combined. Bring to a boil, then reduce the heat to medium-low and simmer, whisking constantly, for 2 minutes, until the mixture thickens. Whisk in the Parmigiano Reggiano, salt, and pepper. Set aside.
Step 3 Combine the flour and ¾ teaspoon salt on a large plate. Whisk the eggs with 2 tablespoons cold water in a wide bowl. Mix the breadcrumbs with ¾ teaspoon salt on a large plate.
Step 4 Lightly coat each slice of eggplant in the flour, shaking off excess. Dip in egg mixture, letting any excess drip off, then dredge in the breadcrumbs, pressing with your fingertips so the crumbs adhere. (Use one hand for the dry ingredients – the flour and breadcrumbs – and one for the egg mixture - it's much neater this way!) Set the breaded eggplant slices on a baking sheet.
Step 5 Line another baking sheet with a few layers of paper towels. Pour enough oil into a large (12-inch) skillet to measure about ¼-inch deep and heat over medium heat until hot (if you dip a piece of eggplant into the oil, it will sizzle immediately). Working in batches, add as much eggplant as will fit in a single layer and fry, flipping once, until golden brown, about 3 minutes per side. Transfer to the paper towel-lined baking sheet. Repeat, frying the remaining eggplant and layering it between sheets of paper towels to drain.
Step 6 Preheat the oven to 425°F and set an oven rack in the middle position.
Step 7 Spread ¾ cup of the marinara sauce over the bottom of a 9x13-inch baking dish. Layer in one-third of the eggplant slices, overlapping the slices to fit. Cover the eggplant with another ¾ cup of marinara sauce. Sprinkle with 1 cup of the mozzarella cheese, then spoon about one-third of the béchamel sauce over the cheese (if the béchamel has solidified a bit, it's fine to add it in dollops; no need to spread it around). Repeat two more layers, leaving the edges of the eggplant exposed on the top layer to create crispy edges. Sprinkle with the Parmigiano Reggiano. Bake until bubbling and golden brown, 30 to 35 minutes. Cool, loosely covered with foil, for about 20 minutes. Scatter the basil over top (if using), cut into slices, and serve.
