In a large Dutch oven, heat butter and olive oil over medium heat until the butter melts.
Add the mushrooms, onion, celery, and garlic. Sauté the vegetables until the onions are translucent, about 5-7 minutes.
Stir in the flour, thyme, salt, and pepper. Cook for 1 minute, stirring constantly.
Gradually add the broth, stirring constantly. Bring to a boil. Once the liquid boils, cook and stir for 2 minutes, or until slightly thickened.
Reduce the heat to low. Add the cooked chicken, cooked rice, and half-and-half or heavy cream. Stir over low heat, just until warmed through (do not boil). Taste and season with additional salt and pepper, if necessary.