Place the eggplant slices in a large bowl and cover with water. Add 1/2 cup of salt and agitate gently to combine. Let stand for 20 minutes, then drain and pat dry with paper towels.

While the eggplant is soaking, heat 1/2 cup of the olive oil in a large skillet and, working in batches, add the potato slices and cook until lightly golden on both sides, 3 to 4 minutes. Transfer to a paper-lined plate to drain and repeat with any remaining potato slices. Season lightly with salt.

Place 1 cup of the flour in a bowl and dredge the eggplant slices in the flour. Shake to remove any excess flour. Add 1/4 cup of the remaining oil to the skillet and, working in batches, fry the eggplant slices until golden on both sides, 2 to 3 minutes per side. Add more oil to the skillet as needed in 1/4 cup increments. Repeat until all eggplant has been cooked. Transfer to a paper-lined plate to drain.

Heat 2 tablespoons of the butter in a large skillet and, when melted, add the onions. Cook until tender, about 4 minutes. Add the ground meats, garlic, and oregano and cook, stirring to break up any lumps, until meat is browned, 4 to 6 minutes. Add the red wine and cook until evaporated. Add the tomato paste and cook, stirring, for 1 minute. Add the crushed tomatoes, water and parsley and cook, stirring occasionally, until almost all liquid is evaporated, 5 to 10 minutes. Season, to taste, with salt and pepper and set aside.

In a medium saucepan melt the remaining 3 tablespoons of butter and, when hot, whisk in the flour and cook, stirring, for 2 to 3 minutes. Gradually whisk in the milk, stirring until smooth and thick. Season with salt, to taste, and whisk in the egg and egg yolk. Stir in 1/2 cup of the cheese and set aside, covered, while you assemble the casserole.

Preheat the oven to 350 degrees F. Lightly grease a 9 by 13-inch casserole dish and place the sliced potatoes in the bottom of the dish in an even layer. Top with half of the eggplant slices and half of the meat sauce. Spoon 3/4 cup of the bechamel sauce evenly over the meat sauce and sprinkle with 1/2 cup of the cheese. Top with the remaining eggplant slices, pressing down on the layers gently with your hands to compress. Top with the remaining meat sauce and then spread the remaining bechamel sauce evenly over the top. Sprinkle with the remaining cheese. Bake, uncovered, until golden brown on the top and heated throughout, 50 minutes to 1 hour. Let stand for at least 40 minutes before cutting into squares and serving.