Place the chicken in a large pot with the carrots, celery, onion, and salt. Add enough cold water to cover (about 8 cups). Bring to a boil. Reduce heat and simmer gently for 45 minutes, skimming off any foam that appears. Add additional water as necessary to keep the chicken covered. Transfer the chicken to a plate and let cool. Remove and discard the carrots, celery, and onion. Add the green beans, asparagus, tomato, and turnip to the broth and simmer until the vegetables are tender, about 5 minutes.
Meanwhile, shred the chicken meat, discarding the skin and bones. Add the meat and pepper to the soup and stir to combine. Ladle into individual bowls.
Tip: For a quicker version of this soup, simmer the vegetables in 8 cups store-bought low-sodium chicken broth until tender, about 5 minutes. Shred the meat from 1 rotisserie chicken and stir it in during the last minute of cooking. Taste the soup for seasoning before adding more salt.