Directions For the Assembly

1. Gather the ingredients for the vegetables.
2. Using a sharp peeler, partially peel the eggplants, leaving strips of peel about 1-inch wide around the eggplant. Slice the eggplant into 1/2-inch slices.
3. Place the eggplant slices in a colander and salt them liberally. Cover them with an inverted plate and weigh down the plate with a heavy can or jar. Place the colander in the sink and let it sit for at least 15 to 20 minutes, preferably 1 hour. 
4. Peel the potatoes, place whole in a pot, cover with cold water, and bring to a boil. Cook until they are just done, about 10 minutes. They should not get too soft, just cooked enough so that they no longer crunch. Drain, cool, and cut them into 1/4-inch slices. Set aside.
5. Heat the oven to 400 F. Line two baking sheets with aluminum foil and lightly grease with oil.
6. Add a splash of water to the egg whites and beat them lightly with a fork. Place breadcrumbs on a flat plate.
7. 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.
8. Place breaded eggplant slices on the foil-lined baking sheets and bake for 30 minutes, turning them over once during cooking.
9. When eggplant is finished cooking, set aside and lower the oven temperature to 350 F.

Make the Meat Filling

1. Gather the ingredients for the meat filling.
2. In a large sauté pan, brown the ground beef until the pink color disappears. Add onion and sauté until translucent, about 5 minutes. Add garlic and cook until fragrant, about 1 minute.
3. Add wine to the pan and allow it to simmer and reduce a bit before adding cinnamon, allspice, parsley, tomato paste, crushed tomatoes, and sugar.
4. Allow the sauce to simmer, uncovered, for approximately 15 minutes so that excess liquid can evaporate. It should be a drier, chunkier sauce. Season to taste with salt and pepper.

Make the Béchamel Sauce

1. Gather the ingredients for the sauce.
2. In a large saucepan, melt butter over low heat. Add flour to melted butter, whisking continuously to make a smooth paste. Let the flour cook for a minute but do not allow it to brown.
3. Add warmed milk to mixture in a steady stream, whisking continuously. Simmer over low heat until it thickens a bit but does not boil.
4. Remove from heat, and stir in beaten egg yolks and pinch of nutmeg. Return to the heat and stir until sauce thickens. Set aside.

Assemble the Moussaka

1. Lightly grease a large deep baking pan (a lasagna pan works perfectly). Sprinkle the bottom of the pan with breadcrumbs.
2. Leaving a 1/4-inch space around the edges of the pan, place the potatoes in a layer on the bottom. Top with a layer of half of the eggplant slices.
3. Add the meat sauce on top of the 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.
4. Pour the béchamel sauce over all, being sure to allow the sauce to fill the sides and corners of the pan. Smooth the béchamel on top with a spatula and sprinkle with remaining grated cheese.
5. Bake for 45 minutes or until béchamel sauce is a nice golden brown color. Allow it to cool for 15 to 20 minutes before slicing and serving.

This has been adapted from thespruceeats.com