To begin, preheat your oven to 400°F (200°C). Place the eggplant slices on a baking sheet and sprinkle both sides lightly with salt. Let sit for 20 minutes to draw out excess moisture and bitterness. Pat dry with a paper towel and brush each slice lightly with olive oil.

Arrange the eggplant slices on a lined baking sheet in a single layer. Roast in the oven for 15-20 minutes, flipping halfway, until they are golden and tender. Remove from the oven and set aside. Lower the oven temperature to 350°F (175°C) for later baking.

While the eggplant is roasting, prepare the meat sauce. In a large skillet over medium heat, warm the olive oil. Add the onion and cook for 5 minutes, until softened. Add the minced garlic and cook for another minute until fragrant.

Add the ground lamb or beef to the skillet, breaking it up with a spoon. Cook until browned, about 5-7 minutes. Drain any excess fat, then stir in the crushed tomatoes, tomato paste, cinnamon, allspice, salt, and pepper. Let the mixture simmer for 15 minutes to allow the flavours to meld. Stir in the parsley, then set the meat sauce aside.

To make the béchamel sauce, melt the butter in a small saucepan over medium heat. Stir in the flour and cook for 1-2 minutes, whisking continuously to avoid lumps and to cook out the raw flour taste.

Gradually whisk in the warm milk, continuing to whisk until the sauce thickens, about 3-4 minutes. Once thickened, remove from heat, stir in the grated cheese, and season with salt and nutmeg. Let it cool slightly, then whisk in the beaten egg to add richness to the sauce.

To assemble, spread a thin layer of the meat sauce in the bottom of a baking dish. Add a layer of roasted eggplant slices, then spread half of the meat sauce over them. Repeat with another layer of eggplant and the remaining meat sauce. Pour the béchamel sauce over the top, spreading it evenly with a spatula.

Bake the moussaka at 350°F (175°C) for 30-35 minutes, or until the top is golden and bubbly. Let it rest for 10 minutes before slicing to allow the layers to set. Serve warm, garnished with a sprinkle of fresh parsley if desired, and accompany with a simple green salad or crusty bread for an authentic Mediterranean experience.