Melt butter in a large skillet over medium heat. Add onion, celery, and garlic; cook until vegetables are softened, about 5 minutes. Sprinkle with flour, salt, thyme, and pepper; cook, stirring constantly, for 1 minute.
Slowly add chicken broth and milk, stirring constantly. Bring to a boil, reduce heat, and simmer for 5 minutes.
Place chicken breasts in a 9x13 inch baking dish. Pour sauce over chicken, then spread cooked rice over the top. Sprinkle with Cheddar cheese.
Bake in preheated oven for 30 minutes, or until chicken is cooked through and cheese is melted and lightly browned.
Interesting Facts
This dish is a great way to use up leftover cooked rice.
You can substitute cooked brown rice for the white rice, if desired.
This dish can be easily doubled to feed a larger crowd.