Melt the butter in a large pot over medium-high heat. Add carrot, celery, onion, green onion, rosemary, thyme, pepper and garlic. Sauté about 8 minutes or until browned.
Stir in broth, scraping pan to loosen browned bits. Stir in turkey and rice; bring to a boil. Cover, reduce heat, and simmer 1 hour and 15 minutes or until rice is tender (15-20 minutes if using quick cooking rice).
Combine flour and milk in a small bowl, stirring with a whisk until there are no lumps. Add to soup and stir well.
Cook over medium heat until thick (about 8 minutes), stirring frequently. Season with salt, to taste.