Place the chicken breasts, wild rice, chicken broth, carrots, celery, onion, garlic, thyme, and bay leaf into the slow cooker.
Add the salt and black pepper, then stir to combine the ingredients evenly.
Cover and cook on low for 6 to 7 hours or on high for 3 to 4 hours until the chicken is fully cooked and the rice is tender.
Remove the chicken breasts and shred them with two forks.
Return the shredded chicken to the slow cooker.
Stir in the heavy cream and let the soup cook for an additional 15 minutes on low to warm through.