Place onion, carrots, and celery in the bottom of a 6-quart slow cooker.Place chicken breasts on top.
Add garlic, rice, chicken broth, bay leaves, Cajun seasoning, salt, pepper, thyme, and rosemary.
Cover and cook on LOW for 4 hours or HIGH for 3.
Add mushrooms and cook 1 more hour.
Just before cooking time is up, melt butter over medium heat in a saucepan. Add flour and cook and stir 1 minute. Whisk in half-and-half. Simmer for 1 minute.
Remove chicken from crock pot and whisk in half-and-half mixture.
Shred chicken and return to slow cooker. Sprinkle with fresh parsley and serve.
Notes
If your chicken breasts are really large, use 2 and cut them in half.