Prepare the Vegetables
Rinse the eggplants and potatoes thoroughly, then slice them into ¼-inch rounds. Pat the eggplants dry with paper towels to remove any excess moisture.
Heat 2 tablespoons of olive oil in a large skillet over medium-high heat. Fry the vegetables in batches until lightly browned, replenishing the oil as needed between batches to prevent sticking.
Transfer the fried vegetables to paper towels to drain off excess oil, season with salt and black pepper, and set aside.
For the Meat Sauce
Warm the olive oil in a large saucepan over medium heat. Add the ground beef and onion, cooking until the beef is fully browned and the onion is softened. Stir in the garlic and cook for an additional 2 minutes.
Incorporate the diced tomatoes, tomato paste, red wine vinegar, cinnamon, oregano, salt, and black pepper. Bring the mixture to a gentle boil while stirring to blend the flavors.
Remove from heat and set aside.
For the Béchamel Sauce
Melt the butter in a medium saucepan over medium-low heat. Add the flour and whisk continuously for about 2 minutes until the mixture thickens and turns lightly golden.
Gradually pour in the warmed milk, whisking constantly to avoid lumps. Increase the heat to medium and continue stirring for 3-4 minutes until the sauce thickens and begins to simmer.
Remove from heat and quickly whisk in the egg yolks, along with salt, black pepper, and nutmeg, ensuring the yolks are fully incorporated without curdling. Set aside.
Assemble the Moussaka
Preheat the oven to 350°F and lightly grease a 9x13-inch baking dish with non-stick cooking spray.
Arrange the potato slices in an even layer on the bottom of the prepared dish. Add half the eggplant slices on top of the potatoes.
Evenly spread the meat sauce over the eggplant layer, then place the remaining eggplant slices on top. Pour the béchamel sauce over the final layer and sprinkle with grated Parmesan cheese.
Bake for 35-45 minutes, or until the top is golden and bubbling. Allow the moussaka to cool for 10 minutes before serving.
Garnish with fresh parsley and serve warm.