Preheat oven to 375°F (190°C). Slice eggplants into 1/2 inch thick rounds and sprinkle with salt. Let sit for 20 minutes to draw out moisture, then pat dry.
Heat olive oil in a skillet over medium heat. Fry eggplant slices until golden brown on both sides. Set aside on paper towels.
In the same skillet, add onion and garlic; cook until softened. Add ground beef and cook until browned.
Add crushed tomatoes, cinnamon, salt, and pepper to the meat mixture. Simmer for 20 minutes.
To make béchamel sauce: melt butter in a saucepan over medium heat. Whisk in flour and cook for 1-2 minutes. Gradually whisk in milk until thickened. Remove from heat and stir in nutmeg, Parmesan, and egg.
In a baking dish, layer half the eggplants, then the meat sauce, then the remaining eggplants. Pour béchamel sauce evenly on top.
Bake for 45 minutes until golden and bubbly. Let cool for 10 minutes before serving.
Notes
This moussaka can be prepared a day ahead and tastes even better after resting overnight.