Spray a 9 X 13-inch baking dish with nonstick cooking spray. Preheat oven to 425 degrees.
Prepare Eggplant: Slice eggplant into 1/2 inch thick rounds. Sprinkle each side lightly with salt and place in a colander set on a plate for 20 minutes. Pat slices with paper towels to remove any excess moisture. Brush 1-2 baking sheets with olive oil. Arrange eggplant slices in a single layer on baking sheets, then brush the tops of slices with a thin layer of olive oil. Roast in preheated oven 15-20 minutes or until the eggplant is soft and beginning to brown. Set aside. Reduce oven temperature to 400 degrees.
Make Tomato Meat Sauce: In a large skillet, heat the butter over medium heat. Add the onion and garlic and cook until they begin to soften about 5 minutes. Add beef or lamb, stirring to break up the meat. Add the allspice, cinnamon, nutmeg, sugar, salt, pepper, and oregano. Cook until the meat is browned and there is no longer any pink in the meat. Drain off any excess fat. Add the wine and cook until it's almost completely evaporated. Add the tomato paste and tomato sauce and cook on low until the sauce has thickened, about 20 minutes. Set aside.
Make Cheese Sauce: Warm the milk in a small saucepan over medium heat until it's simmering. In a medium sauce, melt the butter over medium heat. Whisk in the flour and continue to cook, stirring constantly, for 2 minutes. Whisk in the warm milk, stirring often, and cook until 5-7 minutes until the sauce is smooth and thick. Stir in nutmeg, salt, pepper and Parmesan cheese. Cool sauce for 10 minutes. Slowly whisk in eggs and set aside.
Assemble: Arrange half of the eggplant slices in the prepared baking dish and pour half of the tomato meat sauce on top. Repeat a second layer of eggplant and the remaining tomato meat sauce. Pour the cheese sauce over the tomato meat sauce and sprinkle the remaining Parmesan cheese evenly over the top.
Place baking dish on a rimmed baking sheet. Bake in preheat oven (temperature should be at 400 degrees) for 50-60 minutes and Moussaka bubbly and the top is browned.