Preheat your oven to 400°F (200°C). Arrange eggplant slices on baking sheets, brush with olive oil, and season with salt. Bake for 20 minutes, flipping halfway through, until tender.
In a large skillet, cook the ground lamb over medium heat until browned. Add onion and garlic, cooking until softened.
Stir in crushed tomatoes, red wine, tomato paste, cinnamon, allspice, parsley, salt, and pepper. Simmer for 20 minutes.
For the béchamel sauce, melt butter in a saucepan over medium heat. Stir in flour and cook for 1 minute. Gradually whisk in milk until smooth. Bring to a simmer, stirring constantly, until thickened. Remove from heat, stir in nutmeg, Parmesan, salt, pepper, and egg.
In a baking dish, layer eggplant slices, lamb mixture, and béchamel sauce. Repeat layers, finishing with béchamel sauce.
Bake for 45 minutes, or until golden. Let cool for 20 minutes before serving.