Preheat the oven to 450 degrees.
Peel and slice the eggplants thinly, or slice them unpeeled. Sprinkle the slices generously with salt and allow their bitter juices to drain away in a colander for at least 1/2 hour. Squeeze, wash in cold water, and pat dry. Bake on greased cookie sheets for 40 minutes.
Fry the onions in 2 tablespoons oil until pale golden. Add the ground meat and fry until well browned. Season to taste with salt and pepper, and flavor with allspice or cinnamon if you like. Add the chopped tomato, tomato paste, and parsley. Stir well, moisten with a few tablespoons of water, and simmer for about 15 minutes, or until the meat is well cooked and the water is absorbed.
Put alternate layers of eggplant slices and meat and onion mixture in a deep baking dish, starting and ending with a layer of eggplant.
Reduce the oven temperature to 375 degrees.
Prepare the Bechamel sauce. Melt butter in a saucepan. Add flour and stir over low heat for a few minutes until well blended. Add the hot milk gradually, stirring until it boils, taking care not to allow lumps to form. Season to taste with salt, pepper, and a pinch of nutmeg. Simmer until the sauce thickens. Beat the egg yolk. Stir in a little of the sauce and beat well. Pour back into the pan slowly, stirring constantly. Do not allow the sauce to boil again.
Pour the sauce over the minced meat and eggplant and bake, uncovered, for about 45 minutes, until a brown crust has formed on the top and the layers have fused and blended.
Serve hot, straight from the dish.