Slice the eggplants thinly. Lay them on a baking sheet, brush with olive oil, and sprinkle with salt. Bake for 20-25 minutes, or until tender and lightly browned. Set aside.
For the meat sauce: Heat olive oil in a large skillet over medium-high heat. Add ground lamb and cook, breaking it up with a spoon, until browned. Drain off any excess fat.
Add the chopped onion and cook until softened, about 5 minutes. Add the minced garlic and cook for 1 minute more until fragrant.
Stir in the crushed tomatoes, red wine (if using), oregano, cinnamon, nutmeg, salt, and pepper. Bring to a simmer, then reduce heat and cook for 15-20 minutes, stirring occasionally, until the sauce has thickened.
For the béchamel sauce: Melt the butter in a medium saucepan over medium heat. Whisk in the flour and cook for 1-2 minutes, stirring constantly, until smooth and lightly golden.
Gradually whisk in the warmed milk, a little at a time, until smooth. Bring to a simmer, stirring constantly, until the sauce thickens enough to coat the back of a spoon.
Remove from heat and whisk in the beaten eggs, Parmesan cheese, nutmeg, salt, and pepper until well combined.
To assemble: Spread half of the meat sauce in the bottom of a 9x13 inch baking dish. Layer half of the baked eggplant slices over the meat sauce.
Pour half of the béchamel sauce over the eggplant. Repeat with the remaining meat sauce, eggplant, and béchamel sauce.
Bake for 30-40 minutes, or until the top is golden brown and bubbly. Let stand for 10 minutes before serving.