Spray the inside of a slow cooker with non-stick cooking spray.
Place the chicken breasts in the bottom of the slow cooker in a single layer.
In a medium bowl, combine the stuffing mix, chopped onion, chopped celery, garlic powder, dried thyme, black pepper, and salt.
In a separate bowl, whisk together the condensed cream of mushroom soup, chicken broth, and sour cream until smooth.
Pour the soup mixture evenly over the chicken breasts in the slow cooker.
Sprinkle the stuffing mixture evenly over the top of the soup mixture.
Cover the slow cooker with the lid and cook on low for 4 to 4 1/2 hours, or until the chicken is cooked through and the stuffing is tender.
Carefully remove the chicken breasts and fluff the stuffing with a fork before serving.
Serve the chicken breasts with a generous scoop of the stuffing and sauce.