Fresh lime juice adds extra bright flavor. Bringing the broth back to a boil before serving ensures the snap peas cook to a crisp-tender texture.
Bring first 5 ingredients to a boil in a 3-qt. saucepan over medium heat. Cover, reduce heat to low, and simmer 6 to 8 minutes or until chicken is done. Remove chicken, garlic, and ginger with a slotted spoon, reserving broth in saucepan. Discard garlic and ginger. Let chicken cool slightly (10 to 15 minutes); shred chicken.
Return broth to a boil over medium heat. Add noodles and soy sauce; cook, stirring to separate noodles, 4 to 5 minutes or until just softened. Remove noodles from broth using tongs, and divide among 6 bowls. Place chicken and desired toppings on noodles. Return broth to a boil over medium heat; remove from heat, and stir in lime juice. Divide broth among bowls.