Soak 16 ounces of dried mixed beans (pinto & navy) overnight at room temperature. Drain and sort.
Cook the sausage in a 10-inch skillet over medium-high heat until it's well browned, stirring frequently to separate meat. Pour off fat.
Stir the sausage, broth, onion, carrots, celery, mushrooms, garlic, beans, basil, and black pepper in a 5-quart slow cooker. Cover and cook on LOW for 7 to 8 hours.* Stir in the tomatoes. Cover and cook for 1 hour or until the beans are tender. Sprinkle with the cheese.