First make the tomato sauce: peel and mince onions. Saute onions in about 2 T oil and about 2 T butter, over moderate heat, until they are soft and lightly colored (about 8 minutes). Add mushrooms and saute. Stir in tomatoes, tomato paste, wine, parsley, seasonings and sugar.
Reduce heat to low and simmer uncovered, for 30-45 minutes, stirring occasionally. Most of the liquid should be evaporated, and the mixture quite thick. Remove skillet from heat and let it cool completely.
Start seasoning the eggplants: peel eggplants and slice vertically, about 1/8 to 1/4 inch thick. Sprinkle lightly with salt and let sit for 30 minutes.
Now make the Bechamel Sauce: Place the milk in a saucepan and heat it just until tiny bubbles appear along the edges. Remove and set aside.
Melt one cube butter in a 3-quart saucepan over very low heat until foamy, being careful not to brown. Slowly add the 6 T flour, stirring constantly until smooth (3-4 minutes), and still being careful not to let it brown. Add the milk slowly, whipping with a wire whisk. When the mixture is thick and smooth, remove it from the heat and stir in seasonings. Cool sauce slightly.
Back to the eggplant: Rinse well with cold water; squeeze gently and pat dry. Dredge the eggplant in about 1 1/2 cups flour and saute in olive oil.
Put it all together: Stir ricotta cheese until it is smooth and creamy. Gently fold it into the Bechamel sauce. Stir in beaten eggs until thoroughly incorporated.
Remove and discard any excess oil that has risen to the top of the tomato mixture.
Lightly grease a 16x10 baking pan and sprinkle the bottom with a few bread crumbs. Place a layer of eggplant in the pan, following with a layer of tomato mixture. Sprinkle with bread crumbs and grated cheese. Repeat as many times as you have eggplant to last.
Pour the Bechamel-ricotta sauce over the top and bake at 300 degrees F. for one hour, or until a golden-brown crust has formed on top. Remove moussaka from oven and let it stand undisturbed 20-30 minutes; the delay allows the layers to fuse.
NOTES: Vegetarian version of the Greek classic -- This recipe is not for the faint-hearted. It's very good and it takes a lot of work. Don't waste it on someone who would be just as satisfied with steak and salad!