01 - Preheat oven to 400°F. Line two baking sheets with parchment paper.
02 - Brush eggplant and potato slices on both sides with 2 tablespoons olive oil. Arrange in a single layer on baking sheets. Bake for 20 minutes, flipping halfway, until golden and tender. Set aside.
03 - Heat 2 tablespoons olive oil in a large skillet over medium heat. Add chopped onion and sauté for 4 to 5 minutes until softened. Add minced garlic and cook for 1 minute.
04 - Stir in tomato paste, diced tomatoes, cooked lentils, dried oregano, ground cinnamon, chopped parsley, salt, and black pepper. Simmer for 10 minutes until thickened. Remove from heat.
05 - Melt unsalted butter in a saucepan over medium heat. Whisk in all-purpose flour and cook for 1 minute. Gradually add whole milk, whisking constantly to prevent lumps. Cook until thickened, about 5 minutes. Remove from heat. Stir in ground nutmeg, grated cheese, and a pinch of salt. Allow to cool slightly, then whisk in the large egg.
06 - Reduce oven temperature to 375°F. In a 9x13-inch baking dish, layer half the potato slices, followed by half the eggplant slices, then half the lentil mixture. Repeat layers once more. Pour béchamel sauce evenly over the top.
07 - Bake for 35 to 40 minutes until golden and bubbling. Remove from oven and let rest for 15 minutes before slicing and serving.