Place the chicken breasts in the bottom of the slow cooker. Add the wild rice, carrots, celery, onion, and garlic on top.
Pour the chicken broth over the ingredients. Sprinkle in the thyme, salt, and pepper.
Cover and cook on low for 6 to 7 hours until the chicken is fully cooked and the rice is tender.
Remove the chicken, shred it with two forks, and return it to the slow cooker.
Stir in the half-and-half. Let the soup cook for 15 more minutes on low to warm through before serving.