Preheat the oven to 400°F (200°C).
Arrange the eggplant slices on baking sheets lined with parchment paper. Brush both sides of the eggplant slices with 2 tablespoons of olive oil and sprinkle lightly with salt.
Roast the eggplant in the preheated oven for 20-25 minutes, flipping halfway through, until tender and lightly browned. Remove from oven and set aside.
While the eggplant roasts, 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, breaking it up with a spoon, until browned and cooked through, about 7-8 minutes.
Stir in the crushed tomatoes, tomato paste, dried oregano, ground cinnamon, salt, black pepper, and red wine (if using). Simmer the sauce for 15 minutes, stirring occasionally, until thickened. Remove from heat.
Reduce oven temperature to 350°F (175°C).
To prepare the béchamel sauce, melt the butter in a medium saucepan over medium heat.
Whisk in the flour and cook for 1-2 minutes, stirring constantly, until the mixture is lightly golden.
Gradually whisk in the warmed milk, continuing to whisk until the sauce thickens and comes to a simmer, about 5-7 minutes.
Remove the béchamel from heat and stir in the ground nutmeg and half of the grated Parmesan cheese.
Allow the béchamel to cool slightly, then whisk in the beaten eggs until fully combined.
In a 9×13-inch baking dish, layer half of the roasted eggplant slices evenly on the bottom.
Spread all of the meat sauce evenly over the eggplant layer.
Top the meat sauce with the remaining eggplant slices.
Pour the béchamel sauce evenly over the top layer of eggplant, smoothing it out with a spatula.
Sprinkle the remaining Parmesan cheese over the béchamel.
Bake the moussaka in the preheated oven for 40-45 minutes, until the top is golden and bubbly.
Remove from oven and let rest for 15 minutes before slicing and serving.