This soup has everything you could wish for in a family-friendly dinner recipe—it’s tasty, has vegetables, is easy to make, and it's quick to prepare (only 40 minutes, start to finish).
Combine all ingredients except noodles in a large pot. Bring to a simmer over medium-high heat and simmer until chicken is cooked through, about 12 minutes. Transfer chicken to a plate; when cool enough to handle, shred meat from bones and discard bones. Add meat to soup.
Cook noodles according to package directions, then add to soup. Add more fish sauce, if desired, and serve with lime halves for squeezing.