Preheat your oven to 350 degrees F (175 degrees C). Grease a large casserole dish with cooking spray.
In a large skillet over medium heat, melt the butter. Add the chicken, salt, and pepper and cook until the chicken is cooked through about 10 minutes.
Remove the chicken from the skillet and set it aside.
In the same skillet, add the onion, celery, carrot, mushrooms, thyme, and garlic. Cook until the vegetables are tender, about 10 minutes.
Add the flour to the skillet and stir to combine. Cook for 2-3 minutes, until the mixture thickens.
Slowly add the chicken broth and heavy cream to the skillet, stirring constantly. Cook until the mixture thickens, about 5 minutes.
Add the cooked chicken, cooked rice, and cheese to the skillet, stirring to combine. Pour the mixture into the prepared casserole dish.
Bake the casserole for 25-30 minutes, until the cheese is melted and the casserole is heated through. Let it rest for 5 minutes before serving.
Garnish with chopped parsley or chives, if desired.