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To prepare eggplants: Preheat the oven to 350F/180C. Line two baking trays with parchment paper. Trim and discard the stems and base of the eggplants, and cut each into oval slices about ¼-inch thick. Place them in a single layer on the lined baking trays and brush them on both sides generously with olive oil. Season lightly with salt and pepper. Bake until soft, about 10 - 12 minutes. You’re just looking to soften them, not get them browned, so keep an eye on the oven. Remove and let cool.
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Do the same for the potato slices: Layer in a single layer on lined baking trays (you can use the same trays for the eggplants and potatoes. Brush on both sides with olive oil, season lightly with salt and pepper, and bake until softened but not browned, about 12 - 15 minutes. Remove and cool.
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Prepare meat sauce: In a large heavy skillet, heat 2 tablespoons of olive oil and sauté onion until translucent. Add the ground meat and cook, stirring, until browned. Add garlic, allspice berries, cinnamon, bay leaves, salt, pepper, and stir for a few minutes. Pour in the wine and then the chopped tomatoes. Lower heat, cover, and simmer for 45 to 50 minutes, adding more wine or a little water, if necessary to keep the mixture moist. It should be moist, not liquid. About 5 minutes before turning off heat, add tomato paste. Keep the cover off and simmer until all liquid has been absorbed. Remove skillet from heat and cool slightly.
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While the sauce is simmering, make the bechamél. In a large saucepan, heat the butter over medium heat until melted and bubbling. Be careful not to brown the butter. Add the flour and whisk all together for about 5 minutes, carefully not to brown the roux (butter-flour mixture). The aim here is to cook the flour so that the sauce doesn't end up with a pasty taste. Add the warm milk, whisking all the while. Add the cream and seasonings (salt, pepper, nutmeg). Whisk the bechamél for about 8 - 15 minutes until thick. It should have the consistency of a medium-thick pancake batter. Remove from heat and slide in the eggs, working fast and whisking to combine them without curdling them. Whisk in the cheeses. Cover and set aside until ready to use.
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To assemble the moussaka: Preheat oven to 350° F. Lightly oil an 11x15x3-inch baking pan or oven-proof glass or ceramic dish.
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Mix a ladleful of the bechamél into the meat sauce and stir to combine.
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Spread the slightly cooked potato slices on the bottom of the baking dish. Place an overlapping layer of eggplant slices over bread crumbs or potatoes, patting down with a spatula or your palms. Spread a third of the meat mixture evenly over eggplant. Spread another overlapping layer of eggplant and a third of the meat sauce. Repeat with the last layer of eggplant and finish with a top layer of meat sauce. Pour the remaining bechamel over the meat sauce, spreading it with a spatula so that it is evenly dispersed over the surface. Sprinkle with additional grated cheese and bake for 45 minutes to 1 hour, until bechamél is set and golden brown on top. Remove, cool slightly and serve.