Pour oil into a large Dutch oven. Brown chicken in hot oil over medium-high heat 4 minutes on each side; transfer chicken to a platter.
Sauté onion, carrot, green pepper, and garlic in Dutch oven 5 minutes or until tender. Add mushrooms; cook, stirring often, 3 minutes. Add chicken, chicken broth, and next 8 ingredients. Bring to a boil; cover, reduce heat, and simmer 30 minutes. Add spaghetti, and cook, covered, 12 more minutes. Discard bay leaves. Sprinkle stew with Parmesan cheese. Serve with Garlicky Herb Croutons.