Preheat the oven to 375°F (190°C). Lightly grease a large baking sheet.
Arrange the eggplant slices on the baking sheet, brush both sides with 1 tablespoon of olive oil, and season with salt. Roast for 20 minutes, flipping halfway through, until tender and lightly browned. Remove and set aside.
In a large skillet, heat 2 tablespoons of olive oil 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, breaking it up with a spoon, and cook until browned and no longer pink, about 7 minutes.
Stir in the diced tomatoes, tomato paste, red wine, cinnamon, oregano, salt, and black pepper. Simmer the sauce for 15 minutes, stirring occasionally, until thickened. Remove from heat.
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 until lightly golden and smooth.
Gradually whisk in the warmed milk, continuing to whisk until the sauce thickens and comes to a simmer, about 5 minutes.
Remove from heat and stir in the nutmeg, 1/4 cup of Parmesan cheese, and the beaten eggs. Mix quickly to combine and prevent curdling.
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 with the remaining eggplant slices.
Pour the béchamel sauce evenly over the top layer of eggplant.
Sprinkle the remaining 1/4 cup Parmesan cheese over the béchamel.
Bake uncovered for 30 minutes until the top is golden and bubbly.
Let the moussaka rest for 10 minutes before slicing and serving.