Arrange the sliced eggplant in a large colander, slightly overlapping the slices. Sprinkle each layer lightly with salt.
Continue layering and salting the eggplant until all the slices are in the colander. Leave them for about 30 minutes to release excess moisture.
Meanwhile, begin preparing the meat filling and béchamel sauce.
Preheat the oven to 450°F (240°C).
Pat the eggplant slices thoroughly dry with paper towels. This is important because excess moisture can make the finished casserole watery and overly salty.
Arrange the eggplant on parchment-lined baking sheets. You may need to use multiple trays or work in batches.
Brush the eggplant lightly with olive oil.
Roast for 15–20 minutes, or until the slices are softened and lightly browned. Remove from the oven and set aside.
For the meat filling, heat 1 tablespoon olive oil in a large skillet or heavy pot over high heat.
Add the diced onion and minced garlic. Cook for approximately 2 minutes, stirring frequently.
Add the ground beef or lamb. Cook while breaking it apart with a wooden spoon until the meat is browned and no longer pink.
Pour in the beef broth and red wine vinegar. Stir well and allow it to simmer briefly.
Add the crushed tomatoes, tomato paste, remaining broth as needed, bouillon, bay leaves, sugar, oregano, cinnamon, and salt.
Stir thoroughly and bring the mixture to a simmer.
Reduce the heat to medium-low and cook for about 15 minutes, stirring occasionally, until the filling becomes thick and rich. Remove the bay leaves before assembling.
To make the béchamel, melt the butter in a saucepan over medium heat.
Add the flour and whisk continuously for about 1 minute to cook the flour.
Gradually pour in the milk while whisking constantly to prevent lumps.
Continue cooking and stirring for 3–5 minutes, until the sauce thickly coats the back of a wooden spoon.
Remove the saucepan from the heat.
Whisk in the Parmesan cheese, nutmeg, stock powder or salt, and black pepper.
Allow the sauce to cool for approximately 5 minutes.
Whisk the egg and egg yolk into the slightly cooled béchamel until smooth. Set aside.
Reduce the oven temperature to 350°F (180°C).
Spread half of the roasted eggplant across the bottom of a large baking dish.
Spoon the entire meat filling evenly over the eggplant.
Arrange the remaining roasted eggplant over the meat layer.
Pour the béchamel sauce evenly over the top.
Sprinkle the panko breadcrumbs across the béchamel.
Bake for 30–40 minutes, or until the top is puffed and beautifully golden.
Remove from the oven and allow the Moussaka to rest for at least 10 minutes before slicing.
Garnish with fresh herbs if desired and serve warm.