Greek Moussaka

This is the most delicious Greek moussaka recipe that I have come across, and being Greek born I have tryed many different recipes over the years. I make this often, it is a family favorite at my house, I have even served this many times to guests for a dinner party, everyone loves it, I hope you enjoy it as much as we do!
2 medium eggplants
vegetable oil
1 lb lean ground beef
2 medium onions, peeled and chopped
1 tablespoon fresh minced garlic (or more if desired)
1 (8 ounce) can tomato sauce
1 tsp dried oregano (or to taste)
1 tsp salt, divided (or to taste)
black pepper
2-3 tbsp butter or margarine
2 tbsp flour
1 cup half-and-half cream
1 egg
1/2 cup grated parmesan cheese (can use more)

1. Peel the eggplants, slice 1/4-inch thick.
   
2. Brush cookie sheet with oil.
   
3. Coat each side of sliced eggplant with oil; place on cookie sheet; broil under the broiler until brown; turn and broil the other side, brushing with oil if needed; repeat with all eggplant slices.
   
4. In the bottom of a greased 9 x 9-inch baking pan, arrange half of the eggplant slices.
   
5. In a large skillet, combine beef and onions; cook, stirring until the beef is no longer pink, and the onions are soft; drain fat.
   
6. Add in the garlic, tomato sauce, oregano, 1/2 tsp salt and pepper to taste; pour mixture over eggplant slices.
   
7. Arrange the remaining eggplant slices over the beef mixture.
   
8. PREPARE THE CHEESE SAUCE: Melt the butter in a saucepan, whisk in flour, 1/2 tsp salt and pepper to taste; gradually stir in half and half or milk, cook and stir over medium heat until thick and bubbly.
   
9. In a small bowl, beat egg; stir in some of the hot sauce, then add egg to sauce mixture, mix well; add in Parmesan cheese, and stir again.
   
10. Pour the cheese sauce over mixture in baking dish.
    
11. Bake in a preheated 350 degree oven, for 45 minutes.
    
12. Cut into squares.
    
13. Note: If doubling this recipe (to serve 8), it is best to make two separate casseroles, instead of one larger one.
    
    Dan’s note: also make sure your pan isn’t too big; otherwise everything gets too thin.