Sauté carrots, celery, and mushrooms in unsalted butter until tender.
Add noodles, 3 1/2 cups water, seasoning packets and chicken to vegetables. Cook for 3 minutes.
Combine flour and remaining 1/2 cup cold water. Stir into soup. Cook and stir until thickened, add green onions.
Serve immediately, garnish with cheese and parsley.