Step 1

Make the base and sauce: Preheat the oven to 400°F [200°C] and generously grease two large rimmed baking sheets with about 3 Tbsp of olive oil each. Slice 3 of the eggplants into 1/2‑inch- [12‑mm-] 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, 20-25 minutes. Remove from the oven and set aside. Lower the oven temperature to 350°F [180°C].

Step 2

Meanwhile, chop the remaining eggplant into 1/2‑inch [12‑mm] cubes. Heat the 1/4 cup [60 ml] of oil in a large sauté pan set over medium heat. Add the onions and a pinch of salt and cook, stirring occasionally, until soft and translucent, 5-7 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, 10-15 minutes. Stir in the garlic, oregano, cinnamon, and allspice and cook until fragrant, about 1 minute. Stir in the crushed tomatoes, 1 tsp salt and 1/2 tsp pepper. Turn the heat to medium and simmer until slightly thickened, 5-10 minutes. Taste and add more salt and pepper, if desired.

Step 3

Make the béchamel: 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, 5-10 minutes. Whisk in about half of the Parmesan, 1 tsp 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 [120 ml] of the hot béchamel into the yolks. Turn the heat under the saucepan to medium-low and slowly whisk the tempered egg mixture back into the béchamel. Remove from the heat.

Step 4

Assemble the moussaka: Grease the bottom of a 9-by-13‑inch [23-by-33‑cm] 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 béchamel over the top, smoothing with a rubber spatula. Sprinkle with the remaining Parmesan.

Step 5

Bake until bubbling and golden brown on top, 20-25 minutes. Remove from the oven and let stand for 15 minutes before serving. Serve hot.