Put the wild rice, mushrooms, onion, carrots and celery in a 6-quart slow cooker.
Season the chicken with the salt and thyme, set it over the vegetables and pour in the broth.
Cover and cook on low until the wild rice opens and the thighs are tender, 6 to 7 hours.
Transfer the chicken to a bowl and shred it into bite-size pieces.
Melt the butter in a saucepan, whisk in the flour and cook for 1 minute.
Gradually whisk in the warm milk and simmer until lightly thickened, about 4 minutes.
Stir the milk sauce and chicken into the cooker and heat on high for 10 minutes without boiling hard.