Prep the Vegetables: Using a sharp peeler, partially peel the eggplants, leaving strips of peel about 1 inch wide around the eggplant. Slice the eggplant in to 1/2 inch slices. Place the eggplant slices in a colander and salt them liberally. Cover them with an inverted plate that is weighted down by a heavy can or jar. Place the colander in the sink so that excess moisture can be drawn out. They will need to sit for at least 15-20 minutes, preferably an hour. The salt also helps to remove some of the bitterness of the eggplant. Peel the potatoes and boil them whole until they are just done. They should not get too soft, just cooked enough so that they no longer crunch. Drain, cool and slice them in 1/4 inch slices. Set aside. Preheat the oven to 400 degrees. Line two baking sheets with aluminum foil and lightly grease. Add a splash of water to the egg whites and beat them lightly with a fork. Add breadcrumbs to a flat plate. Rinse the eggplant slices and dry with paper towels. Dip the eggplant slices in the beaten egg whites and then dredge them in the breadcrumbs, coating both sides. Place breaded eggplant slices on baking sheets and bake at 400 degrees for 1/2 an hour, turning them over once during cooking. When eggplant is finished cooking, lower the oven temperature to 350 degrees. Make the Meat Filling: In a large saute pan, brown the ground beef (or lamb) until the pink color disappears. Add onion and saute until translucent, about 5 minutes. Add garlic and cook until fragrant, about 1 minute. Add wine to pan and allow it to simmer and reduce a bit before adding cinnamon, allspice, parsley, tomato paste, crushed tomatoes, and sugar. Allow the sauce to simmer uncovered for approximately 15 minutes so that excess liquid can evaporate. It should be a drier, chunkier, tomato sauce. Season to taste with salt and pepper. Make the Bechamel Sauce: Melt butter over low heat. Using a whisk, add flour to melted butter whisking continuously to make a smooth paste. Allow the flour to cook for a minute but do not allow it to brown. Add warmed milk to mixture in a steady stream, whisking continuously. Simmer over low heat until it thickens a bit but does not boil. Remove from heat, and stir in beaten egg yolks and pinch of nutmeg. Return to heat and stir until sauce thickens. Assemble the Moussaka: Lightly grease a large deep baking pan (lasagna pan). Sprinkle the bottom of pan with breadcrumbs. Leaving a 1/4 inch space around the edges of the pan, place a layer of potatoes on the bottom. Top with a layer of eggplant slices. Add meat sauce on top of eggplant layer and sprinkle with 1/4 of the grated cheese. Top with another layer of eggplant slices and sprinkle once again with 1/4 of the grated cheese. Pour the bechamel sauce over the eggplant and be sure to allow sauce to fill the sides and corners of the pan. Smooth the bechamel on top with a spatula and sprinkle with remaining grated cheese. Bake in a 350-degree oven for 45 minutes or until bechamel sauce is a nice golden brown color. Allow to cool for 15 - 20 minutes before slicing and serving. You can make this dish ahead up until the bechamel sauce and refrigerate. Make the bechamel sauce right before you intend to bake it.