Slice the eggplants approximately 0.5cm thick and spray both sides with olive oil and lightly sprinkle it with salt, leave to rest for 5 minutes. Using a paper towel, gently wipe any excess salt off the eggplants.
Heat up a large frypan on high heat, place in the eggplant slices and cook each side for 3 minutes or until it's a little brown and wilts. Transfer the cooked eggplant slices onto a plate and cook the remaining eggplants and leave aside.
In a large pan or pot heat up the oil on medium to high heat, add in the garlic and onion. Stir for a minute or until fragrant but do not let it burn. Add in the beef, using a wooden spoon break up the mince, add in allspice and cinnamon, cook until beef is brown. Pour in the stock, crushed tomatoes, tomato paste, sugar and oregano and stir to combine. Bring the sauce to a boil then immediately reduce the heat and let it simmer for 15 to 20 minutes or until the sauce reduces and thickens. Taste the sauce and if needed add salt and pepper to taste.
Preheat the oven to 180°C (350°F) or for fan-forced ovens 160°C (300°F). Melt butter in a pan on medium heat, add in the flour and quickly stir for a minute until foamy. Pour in the milk slowly and continue to stir. The mixture will look lumpy but it'll smooth out, just continue to stir quickly. Add in a 1/4 cup of cheese, nutmeg, salt and pepper. Stir until the cheese has melted and the mixture has combined. Remove the Béchamel sauce off the stove and quickly stir in the egg until the mixture is smooth.
Lay half of the cooked eggplants slices in a baking dish that measures approximately 30Lx22Wx7H cm (12x8.5x3 inches). Place the beef sauce filling on top of the eggplants and place the remaining eggplants on top. Pour the cheese sauce over the eggplant and use the back of a wooden spoon to smooth it out. Sprinkle a cup of cheese on top and bake in the oven for 30 minutes. Leave the moussaka to stand for 15 minutes or longer before serving.