Place noodles in a bowl; cover with hot water. Let stand 3 minutes; drain.
Combine 4 cups water and next 6 ingredients (water through 1/2 teaspoon salt) in a Dutch oven; bring to a boil. Add noodles, mushrooms, bell pepper, asparagus, and carrot; bring to a boil. Reduce heat, and simmer 7 minutes.
Sprinkle the fillets with 1/8 teaspoon salt and black pepper. Place fillets on a broiler pan; cook 6 minutes or until fish flakes easily when tested with a fork.
Stir spinach, onion, and cilantro into noodle mixture. Return to a boil; cook 1 minute or until spinach wilts. Ladle soup into serving bowls; top each serving with a fillet.