Salt the eggplant slices on both sides and place in a colander for 30 minutes to draw out moisture. Pat them dry and lay on a lined baking sheet.
1 pound eggplants
Bake eggplant at 425°F (220°C) for 15 minutes, flipping halfway, until golden.
On a separate sheet, bake the potato slices at the same temperature for 15 minutes, also flipping halfway. Lightly season with salt.
2 yellow potatoes
Heat olive oil in a large skillet over medium heat. Sauté onion for 2–3 minutes, then add garlic and cook 2 more minutes. Add ground beef and cook until browned, breaking it apart as it cooks.
2 tablespoons olive oil, ½ cup red onions, 2 cloves garlic, 1 pound ground beef
Stir in sugar, salt, cinnamon, and tomato paste. Cook for 1 minute. Add chopped tomatoes and red wine, scraping the pan to deglaze. Add bay leaf, reduce heat to low, and simmer uncovered for 10 minutes. Discard the bay leaf when done.
1 teaspoon sugar, 1 teaspoon kosher salt, ½ teaspoon ground cinnamon, 3 tablespoons tomato paste, 14 ounces chopped tomatoes, 1 cup dry red wine, 1 bay leaf
In another skillet or saucepan, melt butter over medium heat. Add flour and cook, stirring, for 2 minutes to form a roux. Gradually whisk in the milk. Add salt, pepper, and nutmeg. Simmer until thickened, stirring frequently.
¼ cup butter, ⅓ cup flour, 2 ½ cups whole milk, ¼ teaspoon kosher salt, ¼ teaspoon ground black pepper, ⅛ teaspoon ground nutmeg
Grease a 9×13-inch baking dish with olive oil or butter. Layer the dish by arranging half the potatoes at the bottom, followed by one-third of the eggplant, half the meat sauce, and one-third of the béchamel. Add the remaining potatoes, another third of the eggplant, the rest of the meat sauce, and another third of the béchamel. Finish with the final layer of eggplant, then top with the remaining béchamel and a sprinkle of Parmesan cheese.
¾ cup Parmesan cheese
Bake uncovered for 20 minutes, or until golden and bubbling.
Let cool for 15 minutes before slicing and serving.