Heat the balsamic vinegar in a large, heavy pan. Add the onions, garlic, 1/2 tsp salt and bay leaf and saute 5 minutes. Add eggplant and 1/2 tsp salt, stir and cook, covered 10 minutes, stirring occasionally.
Add mushrooms, green peppers, tomatoes, tomato paste and basil. Mix well and simmer, covered, for 10 minutes. Add Marsala wine, cover and simmer over low heat for about 15-20 minutes. Remove the bay leaf. Serve over pasta or rice.