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Place the vegetables on parchment-lined baking sheets and brush generously with olive oil. Season lightly with salt. Bake at 375F until softened, about 15 minutes for the eggplant and 15 for the potatoes. Remove and set aside. You can do this a day ahead.
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Make the sauce: Heat 3 tablespoons extra virgin Greek olive oil in a large wide pot and cook the onion over medium heat until soft and lightly colored and translucent, about 12 minutes. Stir in the garlic.
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Add the cooked lentils, chopped mushrooms,and chickpeas and stir to coat in the oil. Add the tomato paste and stir. Add the wine, balsamic and tomatoes. Add the bay leaves, cinnamon, allspice and salt and pepper to taste. Cook the mixture for 20 – 25 minutes with the cover ajar on the pot, until the sauce is thick and jammy.
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Remove the bay leaves, allspice and cinnamon. Using an immersion blender, puree about a third of the mixture inside the pot, then mix to combine well. Adjust seasoning with additional salt, pepper, cinnamon or allspice powder and add the dried oregano and mint. Set aside.
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Make the bechamel: Whip the aqua faba in a stand mixer outfitted with a whisk attachment until it looks like meringue. Set aside.
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Heat the four tablespoons of olive oil in a saucepan and add the flour. Stir to make a roux, over medium heat, for about 6 minutes. Add both almond and coconut milks. Whisk while simmering until thick. Cool slightly and fold in the aqua faba. Season to taste with salt, pepper and nutmeg.
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Oil the bottom of a 13 X 9 inch baking pan, preferable glass or ceramic. Place a sprinkling of breadcrumbs on the bottom of the pan. Spread the softened potato slices in one layer. Place a layer of eggplant slices on top. Mix about 1 cup of the béchamel into the bean sauce. Spoon about one-third of the sauce over the eggplant layer. Top with another, final layer of eggplant and spoon over the remaining sauce. Spread it with a spatula or the back of a large spoon so that it is evenly distributed. Ladle and spread the béchamel over the beans evenly.
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Place in a preheated oven at 375F and bake for 45 minutes to an hour, or until the béchamel is set and creamy. Remove, cool slightly and serve.