Preheat the oven to 400°F (200°C). Line two baking sheets with parchment paper.
Arrange the eggplant slices on the baking sheets. Brush both sides with 2 tablespoons of olive oil and season lightly with salt. Roast for 20 minutes, flipping halfway through, until tender and lightly browned. Remove from oven and set aside.
Reduce oven temperature to 350°F (175°C).
Heat 1 tablespoon of olive oil in a large skillet over medium heat. Add the chopped onion and sauté for 5 minutes until softened.
Add the minced garlic and cook for 1 minute until fragrant.
Add the ground beef to the skillet and cook until browned, breaking it up with a spoon, about 7 minutes.
Stir in the crushed tomatoes, tomato paste, oregano, cinnamon, salt, pepper, and white wine. Simmer the sauce for 15 minutes, stirring occasionally, until thickened. Remove from heat.
To make the béchamel sauce, melt the butter in a medium saucepan over medium heat.
Whisk in the flour and cook for 1-2 minutes until it forms a smooth paste without browning.
Gradually whisk in the warmed milk, continuing to whisk until the sauce thickens and comes to a gentle boil, about 5-7 minutes.
Remove from heat and stir in nutmeg, half of the Parmesan cheese, and season with salt and pepper to taste.
In a small bowl, beat the eggs. Slowly whisk about 1 cup of the béchamel sauce into the eggs to temper them, then whisk the egg mixture back into the remaining béchamel sauce.
In a 9×13-inch baking dish, spread a thin layer of meat sauce on the bottom.
Layer half of the roasted eggplant slices over the sauce.
Spread half of the remaining meat sauce over the eggplant.
Add the remaining eggplant slices in a single layer.
Pour the béchamel sauce evenly over the top layer of eggplant.
Sprinkle the remaining Parmesan cheese over the béchamel.
Bake uncovered for 40-45 minutes until the top is golden and bubbly.
Remove from oven and let rest for 15 minutes before serving.
Garnish with chopped fresh parsley before serving.