Lay the eggplant slices in a single layer on a clean kitchen towel or several layers of paper towels. Sprinkle both sides generously with the 1 1/2 teaspoons of kosher salt. Let them sit for 20 minutes, which pulls out bitterness and excess water. After 20 minutes, rinse each slice quickly under cold water and pat them very dry. This step matters a lot. Skipping it will make the finished casserole watery and slightly bitter.

Heat your oven to 400 degrees F. Arrange the dried eggplant slices in a single layer on two lightly oiled sheet pans. Brush the tops with 2 tablespoons of the olive oil and season with a little more salt and pepper. Roast for 20 to 25 minutes, flipping once halfway through, until the slices are tender, golden on both sides, and slightly shrunken. Set them aside. Turn the oven down to 375 degrees F.

While the eggplant roasts, heat the remaining 1 tablespoon of olive oil in a large skillet over medium-high heat. Add the ground lamb and break it up as it cooks. Cook for 6 to 8 minutes, stirring occasionally, until fully browned with no pink remaining. Drain off any excess fat if the lamb is very fatty, leaving about a tablespoon in the pan. Add the onion and cook for 4 to 5 minutes until soft. Add the garlic, tomato paste, cinnamon, allspice, oregano, and black pepper and stir for 1 minute. Add the drained diced tomatoes, stir to combine, and let the mixture simmer on medium-low for 5 minutes until slightly thickened and most of the liquid has cooked off. Taste and adjust salt.

Make the bechamel. Melt the butter in a medium saucepan over medium heat. Whisk in the flour and cook, whisking constantly, for 1 to 2 minutes until the mixture smells slightly nutty and turns a pale golden color. Slowly pour in the warm milk while whisking continuously, adding it in a steady stream to prevent lumps. Keep whisking over medium heat for 4 to 5 minutes until the sauce thickens enough to coat the back of a spoon and holds a line when you drag your finger across it. Remove from heat and stir in the nutmeg, egg yolk, and half of the Parmesan. Season with salt and pepper. The sauce should be noticeably thick at this stage, which is exactly what you want.

Grease your 9x13 inch baking dish lightly with oil or cooking spray. Spread half of the roasted eggplant slices in a single overlapping layer across the bottom of the dish. Spoon all of the lamb filling evenly over the eggplant. Add a second layer using the remaining eggplant slices, overlapping as needed to cover the lamb completely. Pour the bechamel evenly over the top layer, spreading it gently to the edges with a spatula. Sprinkle the remaining Parmesan over the bechamel.

Bake uncovered at 375 degrees F for 30 to 35 minutes, until the bechamel is set, golden brown on top, and the edges are bubbling. The top should feel firm, not jiggly, when you nudge the pan. Let the casserole rest at room temperature for at least 10 minutes before cutting and serving. This resting time is what allows the layers to hold together when you slice and lift portions out of the pan.