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 of the eggplant slices with 2 tablespoons of olive oil and season with salt and pepper.
Roast the eggplant in the oven for 20-25 minutes, flipping halfway through, until golden and tender. Remove from oven and set aside.
Reduce oven temperature to 350°F (175°C).
Heat 1 tablespoon olive oil in a large skillet over medium heat. Add the chopped onion and cook until soft and translucent, about 5 minutes.
Add the minced garlic and cook for 1 minute until fragrant.
Add the ground beef to the skillet. Cook, breaking it apart with a spoon, until browned and cooked through, about 7-8 minutes.
Stir in the crushed tomatoes, tomato paste, oregano, cinnamon, salt, and black pepper. Pour in the white wine and simmer the sauce for 15 minutes, stirring occasionally, until thickened. Remove from heat.
In a medium saucepan, melt 3 tablespoons of butter over medium heat. Whisk in the flour and cook for 1-2 minutes until lightly golden.
Gradually whisk in the warmed milk, stirring constantly to avoid lumps. Cook until the sauce thickens and coats the back of a spoon, about 5-7 minutes.
Remove the béchamel sauce from heat. Stir in the ground nutmeg and half of the grated Parmesan cheese.
Beat the eggs in a small bowl. Slowly whisk about 1/2 cup of the hot béchamel sauce into the eggs to temper them, then whisk the egg mixture back into the saucepan until fully combined.
To assemble the moussaka, spread half of the meat sauce evenly in the bottom of a 9×13 inch baking dish.
Layer half of the roasted eggplant slices over the meat sauce.
Spread the remaining meat sauce over the eggplant layer.
Top with the remaining eggplant slices.
Pour the béchamel sauce evenly over the top layer of eggplant.
Sprinkle the remaining Parmesan cheese evenly over the béchamel.
Bake the moussaka in the preheated oven at 350°F (175°C) for 40-45 minutes until the top is golden and bubbly.
Allow the moussaka to cool for at least 15 minutes before slicing and serving.