Stir the wild rice, white rice and oil in a 3 1/2-quart slow cooker. Cover and cook on HIGH for 15 minutes.
Add the broth, thyme, red pepper, celery, onion and chicken to the cooker. Turn the heat to LOW. Cover and cook for 7 to 8 hours (or on HIGH for 4 to 5 hours) or until the chicken is cooked through.
Serve with the sour cream and green onions, if desired.