Place the chicken breasts in the bottom of the slow cooker. Add the wild rice, mushrooms, onion, carrots, celery, and garlic on top.
Pour the chicken broth over the ingredients. Sprinkle in the thyme, salt, and pepper.
Cover and cook on low for 7 to 8 hours, until the chicken is cooked through and the wild rice is tender.
Remove the chicken breasts, shred them with two forks, and return the meat to the slow cooker.
Stir in the heavy cream until the broth turns creamy and evenly combined. Let the stew sit for 5 minutes before serving.