Preheat the oven to 375°F (190°C).

Slice the eggplants into rounds and sprinkle them with salt. Let them sit for 30 minutes to draw out moisture, then rinse and pat dry.

In a large skillet, heat the olive oil over medium heat. Add the chopped onion and garlic, cooking until softened.

Add the ground beef (or lamb) and cook until browned. Stir in the crushed tomatoes, cinnamon, oregano, and season with salt and pepper. Simmer for 20 minutes.

In a saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1 minute.

Gradually whisk in the milk, stirring constantly until the sauce thickens. Remove from heat and stir in the beaten egg, grated Parmesan, and nutmeg.

In a greased baking dish, layer half of the eggplant slices, followed by the meat sauce, and then the remaining eggplant slices.

Pour the béchamel sauce over the top and sprinkle with additional Parmesan cheese.

Cover the dish with aluminum foil and bake in the preheated oven for 45 minutes.

Remove the foil and bake for an additional 15 minutes until the top is golden brown. Let it cool for 10 minutes before serving.