1. Slice each eggplant lengthwise into 1/4-inch-thick slices. Sprinkle slices generously with salt and let drain in a colander, 20 minutes. Meanwhile, heat 2 tablespoons oil in a large skillet over medium-high heat and saute the onions and garlic until golden, about 10 minutes. Set mixture aside. Cook the ground beef over medium-high heat, breaking it up with a spoon until browned, about 10 minutes. Drain any excess liquid from meat, then add reserved onion mixture, 1 teaspoon salt, and remaining ingredients and simmer for 10 minutes. Dry the eggplant slices on paper towels. Starting with 2 tablespoons olive oil in a nonstick skillet over medium heat saute eggplant until lightly brown on both sides. Transfer slices to paper towels. Repeat with remaining eggplant adding olive oil as needed.Make bechamel sauce::
2. In a medium heavy saucepan over low heat melt the butter. Stir in the flour and cook, uncovered over very low heat, 2 to 3 minutes. Add a small amount of the milk and stir until thoroughly mixed. Stir in the salt, white pepper, and nutmeg. Continue adding milk very slowly, stirring well. Keep stirring until the mixture becomes the consistency of light cream, 8 to 10 minutes. Slowly stir in the half-and-half cream. In a medium bowl beat together eggs and egg yolks. Add a small amount of the hot cream mixture to the eggs and whisk well. Slowly add the egg mixture to the milk mixture, stirring constantly over low heat, 5 minutes more (sauce should be the consistency of thick cream).
3. To assemble, place a layer of eggplant in the bottom of a 13x9-inch baking dish. Spread the meat mixture over eggplant and top with another layer of eggplant. Spread the bechamel sauce over top and sprinkle with Parmesan cheese. Bake on a cookie sheet until the top is golden brown, 30 to 40 minutes. Let casserole stand 30 minutes before slicing. Makes 8 servings.