Wash the eggplants well. Slice them into even rounds. Place them on a baking tray. Brush both sides with olive oil. Sprinkle with salt and pepper. Roast in a 400°F oven for about 20 minutes until soft and lightly golden.

Heat olive oil in a large skillet. Add diced onion and cook for about five minutes. Stir often until soft. Add minced garlic and cook for one more minute. The kitchen will start smelling amazing.

Add the ground lamb or beef to the skillet. Break it apart with a spoon. Cook until browned completely. Drain any extra fat if needed. This step gives the moussaka its rich and hearty texture.

Add crushed tomatoes, tomato paste, oregano, thyme, cinnamon, salt, and pepper. Stir everything together. Reduce the heat and let it simmer for 20 minutes. The sauce should become thick and flavorful.

Melt butter in a saucepan over medium heat. Add flour and whisk constantly for one minute. Slowly pour in the milk while whisking. Continue stirring until smooth and thick. Add nutmeg, salt, pepper, and Parmesan cheese.

Remove the saucepan from heat. Let it cool slightly. Slowly stir in the beaten egg. Mix well until smooth. The sauce should be creamy and rich without any lumps.

Place half of the roasted eggplant slices in the baking dish. Spread the meat sauce evenly over them. Add the remaining eggplant slices on top. Pour the béchamel sauce over everything and smooth the surface.

Bake at 375°F for about 40 minutes. The top should be golden brown and slightly firm. Let the dish rest for 15 minutes before slicing. This helps the layers stay together.