Rinse the rice under running water. Place the uncooked rice, chicken breast, garlic, onions, celery, carrots, thyme, rosemary, salt, bay leaves, and chicken broth into your slow cooker.
Cover and cook on the high setting for 3-4 hours or on the low setting for 7-8 hours.
In the last 1/2 hour of cooking, remove the chicken from the slow cooker. Allow to cool slightly and then use two forks to shred the chicken into small pieces.
When the rice is done cooking, remove the bay leaves and add the shredded chicken back into the slow cooker.
Melt the butter or oil in a saucepan. Add the flour and let the mixture cook for 1 minute. Whisk the mixture slowly while adding in the milk. Continue to whisk until all lumps have dissolved. Allow the mixture to thicken and become creamy.
Add the creamy mixture into the slow cooker. Stir to combine. Add additional broth to your preference if the consistency is too thick. Taste and season with salt and pepper if needed. Garnish with fresh thyme and serve.