Slice eggplant and sprinkle with salt; set aside for 15 to 20 minutes. Pare and slice potatoes; dry eggplant slices. Brown eggplant and potato slices in olive oil. Drain on paper towels and set aside. Sauté onion and ground beef in 2 tablespoons melted butter until brown; add parsley and tomato sauce and simmer for 10 minutes.
Dust a greased 13x9x2-inch baking dish with breadcrumbs. Place a layer of potatoes, eggplant, and meat mixture in pan; repeat layers, ending with eggplant. Combine remaining butter and flour; cook over low heat, stirring constantly. Heat milk; gradually add to flour mixture, stirring constantly. Cook over low heat until thick; cool slightly.
Beat egg yolks, and combine with cheese and nutmeg; add to sauce. Beat egg whites until stiff and fold into sauce. Pour sauce over eggplant-meat layers. Bake at 350° for 45 to 55 minutes or until golden brown.Serves 12.