In a large sauce pan, saute the onions, garlic in 3 tablespoons of butter. Add in the beef and stir until golden brown, then gradually add in the wine. Toss in the tomatoes, water, salt and pepper, parsley, oregano and sugar. Let this boil for about 1/2 hour. When the beef sauce is done take it off the stove and add in the beaten eggs, the breadcrumbs and 1/2 cup of the shredded cheese. Set it a side.
In a large frying pan add in the 1/2 cup Olive oil or Vegetable Oil , and fry the eggplant and potatoes until lightly brown. Take a deep buttered baking pan and start layering in the potatoes, (the potatoes actually cover the bottom of your pan), and then continue layering the eggplant slices on top. Add your beef sauce on top of the eggplants and spread throughout the pan.
In a large sauce pan, melt butter on low heat with a whisk and slowly add 1 tablespoon of flour at a time, gradually add the milk, whisk continuously . Add in all the other ingredients and stir until your Bechamel sauce has thicken. Pour on top of the meat sauce and spread.
Bake at 180 degrees for 1 hour and until Bechamel sauce is golden brown.