Using a 4- to 5-quart slow cooker, combine onion, green beans, shallot and chicken. Sprinkle with rosemary, thyme, salt and pepper. Add chicken broth and vinegar to mixture in cooker.
Cover and cook on low-heat setting for 6 to 7 hours or on high-heat setting for 3 to 3 1/2 hours.
Remove chicken and vegetables to a serving platter, reserving cooking liquid. Cover chicken and vegetables with foil to keep warm.
If using low-heat setting, turn to high-heat setting. Combine in a small bowl cornstarch and water. Stir into liquid in cooker. Cover and cook about 15 minutes more or until thickened. Spoon sauce over chicken and vegetables.