Heat 2 tablespoons of the oil in a heavy skillet, then add onions and cook until soft and transparent. Blend in the chopped meat and break up with a spatula until it browns thoroughly. Simmer a few minutes stirring constantly. Add the wine, tomatoes and herbs, then season with salt and pepper, cover and simmer for 30 minutes. Should be a little thick. Cool. Add the egg whites and 2 tablespoons or more of the bread crumbs and blend thoroughly with a spoon.
Partially peel the eggplants by cutting 1/2-inch strips of skin off lengthwise, cut off the green stem end, then cut in circles or lengthwise into 1/3-inch slices. Salt, and let stand for a while in a colander with a weight on the slices for about 30 minutes in order to get the bitterness out of the eggplants. Rinse, and dry the eggplant slices, then fry lightly in batches on both sides in the remaining hot olive oil, adding more if necessary (or skip the frying of eggplant slices and broil them in the oven with a little olive oil or Pam smeared on them and on the broiler pan, until slices are softened and brown. This saves some extra fat in this dish)
If using the frying method, drain on paper towels when each batch has finished browning. Eggplant takes in a lot of oil using this method.
Prepare the white sauce and cool slightly, then add the lightly beaten egg yolks tempering the white sauce with a little of the beaten egg yolks if necessary prior to adding the rest of the egg mixture, about half of the grated cheese, nutmeg, some white pepper if you have it, and stir well.
Butter a 9 x 12 x 2-inch baking pan and sprinkle lightly with the bread crumbs. Line the bottom of the pan with half the eggplant slices, then cover completely with the meat mixture, spreading evenly with a spatula. Layer the remaining eggplant slices over the meat. Cover with the sauce and sprinkle with the remaining cheese, nutmeg, and remaining bread crumbs. Add small pieces of butter here and there over the bread crumbs. (Can make more than one layer of meat and eggplant slices if you with. I have also added round blanched partially cooked slices of potatoes or you could also use zucchini squash slices with the eggplant prepared like the eggplant slices, fried or broiled or all three vegetables together.)
Bake in a moderate oven (350 degrees) for about 35 to 40 minutes, or until the top turns golden in color. Remove from the oven to a rack and allow to cool for at least 10 minutes, then cut in squares, transfer to a warm serving platter, and serve warm. (Is also good cold or at room temperature)