Combine noodles, oil, and 1/2 teaspoon salt; toss well to coat.
Combine water and next 6 ingredients (through chicken breast) in a large Dutch oven; add 1/2 teaspoon salt. Bring to a boil. Cover, reduce heat, and simmer 30 minutes. Remove chicken from pan; cool slightly. Shred chicken with 2 forks.
Add carrot and mushrooms to pan; bring to a boil. Reduce heat, and simmer 6 minutes or until carrot is tender. Add shredded chicken, parsley, spinach, and remaining 1 teaspoon salt; cook 3 minutes or until spinach wilts. Stir in noodle mixture, lemon juice, and pepper. Cook 1 minute.