In a large stockpot over medium-high heat, saute carrots, onion, celery, and garlic in butter until just tender, about 5 minutes. Quickly stir in flour until combined then slowly stir in broth, a little at a time, until mixture is smooth and lump free. Add wild rice and turkey to broth and bring soup to a simmer, stirring occasionally until vegetables are tender, about 10 minutes.