Heat 1 tablespoon of olive oil in a large skillet over medium-high heat. Stir in the ground beef and onion, and cook until the beef is browned.
Drain off any excess oil, then stir in the oregano, parsley, tomato paste, sugar, salt, and pepper, add 1/2 -3/4 cup of water.
Bring to a boil on high heat then lower heat to simmer for 1-2 hours or at least 30 minutes. I find that cooking this for about 2 hours gives it more flavor and umami, that's why I always do it ahead of time.
Preheat and lightly oil indoor (or outdoor grill). Wash and peel eggplant, cut thinly and brush with olive oil (or lightly spray with cooking oil) then season with salt and pepper. Grill eggplant until just tender.
Preheat oven to 375 degrees F (190 degrees C).
Melt butter in a large saucepan over medium heat. Whisk in the flour, and cook until the flour is slightly toasted, about 5 minutes. Whisk in milk, nutmeg, and salt.
Bring to a simmer over medium-high heat, then reduce heat to medium-low and simmer for 10 minutes more; add in grated cheese and cook until cheese is completely melted.
Lay grilled eggplant slices on the bottom of the pan, spread half of the meat sauce over th e eggplant.
Repeat layer and pour the white sauce over top, and smooth with a spatula.
Sprinkle grated quickmelt cheese or mozzarella on top.
Bake moussaka in preheated oven until bubbly and golden brown, about 30 minutes.
Do the meat sauce first. This can be made ahead of time like a day before and just reheat it.