Wash the eggplants and cut off the ends. With a small paring knife, working with one eggplant at a time, peel 1/2-inch strips of the skin from the stem down, making stripes around the eggplant. Place it on its side and carefully slice crosswise into even, 1/3-inch-thick rounds. Repeat with the remaining eggplants. Sparingly, use 2 teaspoons salt to very lightly sprinkle both sides of all the slices, spreading the salt with your fingers. Don't use more than 2 teaspoons of salt--a little salt goes a long way. Place the salted slices on a large flat tray lined with paper towels. Layer them between additional towels if necessary and let them sit for approximately 30 minutes. Rinse the slices well under cold running water and immediately pat dry and set aside.
In a very large saute pan, heat 2 tablespoons of the olive oil over medium-high heat. Make sure the oil is hot and evenly distributed over the surface of the pan. Carefully add 6 to 8 eggplant slices (or as many as the pan will fit comfortably), one at a time, and fry, turning once, until the eggplant is golden brown on both sides, about 3 to 4 minutes per side. Using tongs, remove the eggplant from the pan and place on a tray lined with paper towels. Continue frying the remaining eggplant in batches, repeating the process with 2 tablespoons of oil each time until all the eggplant has been browned, drained, and cooled (see note).
To make the meat sauce, in a large skillet, melt 2 tablespoons of butter over medium heat, being careful not to brown it. Add the chopped onions and cook until tender and translucent, 3 to 4 minutes. Add the ground beef and brown for about 6 minutes. Add the wine and scrape the bottom of the pan with a spatula or spoon to deglaze, loosening any brown bits stuck on the bottom. Bring to a simmer and cook to reduce the liquid until the pan is almost dry. Add the tomato paste, water, parsley, cinnamon, salt, and pepper and let the mixture cook over very low heat for about 20 minutes. The sauce should be moist and spreadable but it will have much less liquid than a standard pasta sauce. Remove from the heat and let cool.
To make the béchamel sauce, in a saucepan, melt 4 tablespoons of butter over medium-high heat. Blend in the flour with a whisk and cook until bubbly but not brown, 2 to 3 minutes. Gradually pour in the milk, whisking it into the butter-and-flour roux thoroughly until the mixture is thick and smooth. Remove the mixture from the heat and add the nutmeg, salt, and pepper to taste and blend thoroughly. Let the mixture cool slightly, then add 1 cup of the grated cheese, stirring until blended.
Preheat the oven to 350F.
Before assembling the moussaka, butter a 9-by-13-by-2 inch baking dish. Arrange half of the eggplant on the bottom of the pan and spread half of the meat mixture over it. Repeat with another layer of eggplant and top with the remaining meat mixture. Spread the béchamel sauce evenly over the meat mixture, and then sprinkle the remaining 1 cup cheese over the top. Bake for 40 minutes or until casserole is piping hot and the cheese has melted and browned slightly. Transfer to a wire rack and let cool for about 20 minutes. Cut into 3-inch squares and serve warm.
Cat's Notes:
Eggplant acts as a sponge when it comes in contact with oil, so keep the oil to a minimum and make sure it is hot and evenly distributed in the pan. Salting the eggplant before sauteing also helps the eggplant absorb less oil.
