1. Make the base and sauce: Preheat the oven to 400 F. and generously grease 2 large-rimmed baking sheets with about 3 tablespoons of olive oil each. Slice 3 of the eggplants into 1/2-inch-thick rounds. Arrange the eggplant slices in a single layer on the sheets, brush the tops generously with more oil, and season with salt and pepper. Bake, flipping the eggplant pieces once, until softened and lightly browned, 20 to 25 minutes. Remove from the oven and set aside. Lower the oven temperature to 350 F.
2. Meanwhile, chop the remaining eggplant into 1/2‑inch cubes. Heat the 1/4 cup of oil in a large saute pan set over medium heat. Add the onions and a pinch of salt and cook, stirring occasionally, until soft and translucent, 5 to 7 minutes. Add the chopped eggplant and mushrooms, turn the heat to medium-high, and cook, stirring occasionally, until the vegetables are very tender and the liquid has cooked off, 10 to 15 minutes. Stir in the garlic, oregano, cinnamon and allspice, and cook until fragrant, about 1 minute. Stir in the crushed tomatoes, 1 teaspoon salt and 1/2 teaspoon pepper. Turn the heat to medium and simmer until slightly thickened, 5 to 10 minutes. Taste and add more salt and pepper, if desired.
3. Make the bechamel: Melt the butter in a medium saucepan set over medium heat. Add the flour and whisk until fully combined, then slowly whisk in the milk. Simmer, whisking constantly, until the sauce thickens enough to coat the back of a spoon, 5 to 10 minutes. Whisk in about half of the Parmesan, 1 teaspoon salt and a generous amount of pepper. In a separate small bowl, whisk the egg yolks until smooth. Whisking constantly, slowly drizzle about 1/2 cup of the hot bechamel into the yolks. Turn the heat under the saucepan to medium-low and slowly whisk the tempered egg mixture back into the bechamel. Remove from the heat.
4. Assemble the moussaka: Grease the bottom of a 9-by-13‑inch baking dish. Layer half of the eggplant slices in the bottom of the baking dish and cover with half of the sauce. Layer the remaining eggplant slices into the dish and cover with the remaining sauce. Evenly pour the bechamel over the top, smoothing with a rubber spatula. Sprinkle with the remaining Parmesan.
5. Bake until bubbling and golden brown on top, 20 to 25 minutes. Remove from the oven and let stand for 15 minutes before serving. Serve hot. Store leftovers, covered, in the fridge for up to 3 days. Serves 8-10.
(Leah Koenig's writing and recipes have appeared in The New York Times, The Wall Street Journal, Saveur, Epicurious, Food52, BonAppetit.com, Departures, More and Tablet, among other publications. Her most recent cookbook, “Little Book of Jewish Appetizers [Chronicle Books], came out in August 2017. Her previous book, “Modern Jewish Cooking” [Chronicle], was published in 2015 and featured in Food52's Piglet cookbook tournament.)
The Nosher food blog offers a dazzling array of new and classic Jewish recipes and food news, from Europe to Yemen, from challah to shakshuka and beyond. Check it out at www.TheNosher.com.