Combine first 5 ingredients in a Dutch oven. Bring to a boil; cover, reduce heat, and simmer 45 minutes or until chicken is done. Remove chicken from broth, and set aside; remove and discard celery, onion, and carrot. Skim fat from broth. Add water, if necessary, to broth to equal 8 cups. Set aside 2 cups broth. Bone chicken; cut into bite-size pieces.
Bring broth in Dutch oven to a boil; add spaghetti. Cook 10 minutes. Drain.
Coat a nonstick skillet with cooking spray; place over medium-high heat until hot. Add mushrooms and next 4 ingredients; saute until tender. Add reserved 2 cups broth, tomatoes, and next 6 ingredients. Bring to a boil; reduce heat, and simmer, uncovered, 40 minutes. Add tomato mixture and cheese to spaghetti; toss. Spoon into 3 (1-quart) casseroles coated with cooking spray. Cover and bake at 350° for 30 minutes.