Slice the eggplants into 0.6 cm slabs, discarding the outermost skin slices. Sprinkle with salt and let rest for 30 minutes to remove bitterness.
Heat 2 tablespoons olive oil in a large skillet over medium heat. Add chopped onion and cook until golden, about 5 minutes.
Add ground lamb or beef and cook until browned. Drain excess fat and return skillet to heat.
Stir in oregano, cinnamon, black pepper, nutmeg, and paprika. Pour in red wine and boil for 1 minute to reduce.
Add diced tomatoes, sugar, and hot beef broth. Simmer on medium-low until thickened and aromatic, about 20 to 30 minutes.
Preheat broiler to high. Pat eggplant slices dry and brush both sides with olive oil.
Arrange eggplant slices on an oiled sheet pan in one layer. Broil about 15 cm from heat, turning once, until golden and tender. Work in batches if needed.
Turn off broiler and set oven to 175°C (350°F). Lightly oil a 24 x 33 cm baking dish.
Heat remaining olive oil in a saucepan over medium-high heat. Whisk in flour, salt, and pepper, cooking until golden.
Gradually whisk warm milk into the flour mixture, cooking until thick enough to coat a spoon, about 5 to 7 minutes. Stir in nutmeg.
Whisk eggs in a separate bowl. Temper eggs by slowly whisking in a ladle of hot bechamel, then combine egg mixture back into bechamel. Cook gently for 2 minutes until creamy.
Layer half the eggplant slices in the prepared dish. Spread all meat sauce evenly over eggplant.
Top with remaining eggplant slices. Pour bechamel sauce over the surface and spread evenly. Sprinkle breadcrumbs on top.
Bake at 175°C (350°F) for 45 minutes on the center rack until set and golden. Optionally broil briefly for a deeper crust, watching closely.
Let rest for 10 minutes before slicing. Serve warm.