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