Cut each eggplant lengthwise into 3 equal slices. Sprinkle with the salt and let stand for 20 minutes. Dry the slices on paper towels to absorb the liquid that accumulates. Set aside.
Heat 1 tablespoon of the oil in a pan and stir-fry the garlic over moderate heat until golden, about 2 minutes. Add the tomato sauce, tomato paste, and water and mix well. Simmer the sauce over low heat until it becomes thickened, about 15 minutes.
Heat the remaining 3 tablespoons of oil in a skillet. Dip the eggplant slices into the beaten eggs and fry over moderate heat for 2 minutes. Fry all the eggplant slices this way.
Assemble the moussaka in this manner: Put a layer of eggplant in a heatproof glass or metal baking dish. Cover with about half cup of the cooked tomato sauce. Sprinkle with about 2 tablespoons of the grated cheese. Cover this with another layer of eggplant, then sauce, then cheese. The last layer (there should be three) ends with the cheese.
Bake in a preheated 350 degree oven for half hour.
Serve warm. Serves 6 with spaghetti or other dishes.