Heat oil in 12 inch fry pan over medium heat. Add onion, cooking til soft, about 7 minutes. Add garlic and cook 1 more minute. Add mushrooms, bell pepper and eggplant. Cook, stirring, until liquid has evaporated, about 5 minutes.
In a 3 or 4 quart casserole, combine eggplant mixture with tomato sauce, broth, thyme and rice. Cover and bake till rice is tender (about 1 hour).