In a 3 1/2- to 5-quart slow cooker place mushrooms, onion, and garlic. Add chicken. Combine in a medium bowl the chicken broth, tomato paste, wine, basil, salt, and pepper. Pour over chicken in cooker.
Cover and cook on low-heat setting for 5 to 6 hours or on high-heat setting for 2 1/2 to 3 hours.
Remove chicken to a serving platter, reserving cooking liquid. Cover chicken with foil to keep warm.
If using low-heat setting, turn to high-heat setting. Combine in a small bowl the cornstarch and water. Stir into liquid in cooker. Cover and cook for 15 minutes more or until thickened. Spoon sauce over chicken. Sprinkle with Parmesan cheese.