Lay the slices of eggplant on paper towels, sprinkle lightly with salt, and set aside for 30 minutes to draw out the moisture. Then heat the olive oil in a frying pan and fry the eggplant until browned. Use more oil if you need. Drain on paper towels.
While tha eggplant is being salted, in a large frying pan over medium heat, melt the butter and add the minced lamb, salt and pepper to taste, onions, and garlic. After the lamb is browned, add the cinnamon, nutmeg, oregano and parsley. Pour in the tomatoes and wine, and mix well. Simmer for 20 minutes. Allow to cool, and then stir in beaten egg.
To make the white sauce, begin by heating the milk in a saucepan. Melt the butter in a separate saucepan over medium heat. Whisk in flour until smooth. Lower heat; gradually pour in the hot milk, whisking constantly until it thickens. Season with salt, and white pepper.
Arrange a layer of eggplant in a greased 22x33cm baking dish. Cover eggplant with all of the meat mixture, and then sprinkle 1/2 cup of Parmesan cheese over the meat. Cover with remaining eggplant, and sprinkle another 1/2 cup of cheese on top. Pour the white sauce over the top, and sprinkle with the nutmeg. Sprinkle with the remaining cheese.