Sprinkle the diced eggplant with salt and let it stand in a bowl for 10 minutes to draw out the bitterness. Rinse under cold water, drain and pat dry.
Preheat oven to 375°
Heat ¼ cup of the olive oil in a large cast-iron frying pan. Sauté the eggplant over a low flame, adding more oil as you need it, for about 20 minutes, or until tender and golden. (If the pan is not large enough, you may have to do this in two batches.) Drain the eggplant on paper towels and put them in an oiled baking dish. Put 2 tablespoons of the olive oil in the frying pan and cook the onion slowly until transparent. Add the meat, tomatoes, and parsley and cook for 5 minutes. Turn off the heat. Add the garlic and stir well. Spoon the mixture over the eggplant and sprinkle with cheese, bread crumbs, and the remaining olive oil. Bake for 30 minutes.
Serve sprinkled with basil or parsley and with a bowl of warm coulis sauce.