Season the chicken with the salt and pepper and place it in a 6-quart slow cooker.
Add the wild rice, onion, carrots, celery, mushrooms, broth, thyme and bay leaf.
Cover and cook on low until the rice grains split and the chicken reaches at least 175°F, 6 to 7 hours.
Transfer the chicken to a plate, pull it into bite-size pieces and discard the bay leaf.
Melt the butter in a saucepan over medium heat, whisk in the flour for 1 minute, then slowly whisk in the milk.
Cook the milk mixture until it coats a spoon, 4 to 5 minutes, then stir it and the chicken into the cooker.
Cover and cook on high for 10 minutes, then thin with hot broth if the rice has made the soup too thick.