Place the chicken breasts, rinsed wild rice, carrots, celery, onion, and minced garlic into the slow cooker insert.
Pour the chicken broth over the ingredients and add the thyme, rosemary, salt, and pepper.
Cover and cook on High for 3.5 to 4 hours or on Low for 6 to 7 hours, until the rice is tender and the chicken is fully cooked.
Remove the chicken breasts from the slow cooker and place them on a cutting board; shred the meat using two forks.
Return the shredded chicken to the slow cooker.
Stir in the heavy cream and the fresh baby spinach.
Cover and cook for an additional 15 minutes until the spinach is wilted and the soup is heated through.
Taste and adjust seasoning with additional salt or pepper if necessary before serving.