Classic Greek Eggplant Moussaka

Name: Classic Greek Eggplant Moussaka
Ingredients: 3 to 4 eggplants (about 4 pounds); Salt (for salting the eggplants); 1 pound potatoes (peeled, boiled); Splash of water; 8 large egg (whites and yolks separated); 1/2 cup olive oil (divided); 2 cups plain breadcrumbs (divided); For the Meat Filling; 1 1/2 pounds ground beef (or lamb); 2 large onions (finely diced); 2 cloves garlic (minced); 1/2 cup dry red wine; 1 teaspoon ground cinnamon; 1/4 teaspoon ground allspice; 1/4 cup fresh parsley (chopped); 2 tablespoons tomato paste; 1 cup crushed tomatoes; 1 teaspoon sugar; Salt (to taste); Freshly ground pepper (to taste); For the Béchamel Sauce; 1 cup (2 sticks) unsalted butter; 1 cup all-purpose flour; 4 cups milk; Pinch of ground nutmeg; For the Assembly; 1 cup Kefalotyri cheese (grated, or freshly grated Parmesan cheese)