Place the chicken breasts in the bottom of the slow cooker. Add the rinsed wild rice, sliced carrots, sliced celery, diced onion, minced garlic, dried thyme, salt, and black pepper.
Pour the chicken broth over the ingredients. Stir gently to combine everything except the chicken, which should stay at the bottom.
Cover and cook on low for 6 to 7 hours, until the rice is tender and the chicken reaches an internal temperature of 165 degrees.
Remove the chicken breasts, shred them with two forks, and return the meat to the slow cooker.
Stir in the heavy cream. Cover and cook on low for 20 to 30 minutes more, until the broth is heated through and creamy.