In a large stockpot or Dutch oven set over medium heat, add the butter. Once melted, add the celery and carrots. Saute for 3 minutes. Add the mushrooms and cook until browned, about 3 minutes. Add onion and garlic, continue cooking for an additional 2 minutes, stirring frequently.
Pour in broth and add salt, pepper, and seasonings/herbs. Turn heat up and bring to a boil. Once boiling, add rice and cook covered, for 12 minutes. Remove lid and add turkey and thyme. Continue cooking for an additional 5 minutes or until rice is tender.
Remove from the heat and serve immediately.