Melt butter in a 4- to 6-gallon stockpot over medium heat; add onions and celery, and sauté until tender.
Add flour, stirring until blended; cook, stirring constantly, 2 minutes. Stir in broth and milk, stirring constantly. Bring to a boil, stirring constantly, and cook 2 minutes. Stir in chicken, sliced carrots, and remaining ingredients.