Place the chicken breasts into an even layer in the slow cooker.
Add rinsed rice, celery, carrots, onion, and garlic on top.
Pour the chicken broth inside with salt, pepper, dried thyme, and bay leaves.
Place lid on top and cook on the low setting for 6 hours. In the last 30 minutes of cooking, add the mushrooms.
Discard the bay leaves. Remove the chicken breasts and shred using two forks. Return shredded chicken to slow cooker and stir into the soup.
Melt butter in a saucepan over medium heat. Stir in the flour and cook for 1 minute, stirring frequently until there are no more lumps. Pour in the milk and whisk it in until smooth and thickened. Stir into the soup until well combined.