Combine noodles, oil, and 1/2 tsp salt; toss well to coat.
Combine water and next 5 ingredients in a large Dutch oven; add 1/2 tsp salt. Bring to a boil. Cover, reduce heat, and simmer for 30 minutes. Remove chicken from pan; cool slightly. Shred chicken with 2 forks.
Add carrot and mushrooms to pan; bring to boil. Reduce heat and simmer 15 minutes or until carrot is tender. Add shredded chicken, parsley, spinach, and remaining 1 teaspoons salt. Cook 5 minutes or until spinach is wilted.
Stir in noodles, lemon juice and pepper. Cook for an additional minute. Best made 1 day ahead.