Bring chicken and water to a boil: Bring chicken, water, and 1 tablespoon salt to a boil in a large stockpot.
Add onions, celery, and garlic and simmer: Skim foam. Add onions, celery, and garlic. Reduce heat. Simmer, partially covered, for 30 minutes.
Remove breast, add carrots, and cook: Remove chicken breast, and set aside. Add carrots. Simmer, partially covered, for 40 minutes.
Remove rest of chicken and cut meat into pieces: Remove remaining chicken; discard back and wings. Let cool slightly. Remove meat from bones, and cut into bite-size pieces.
Add chicken to soup and season with salt: Stir in desired amount of chicken; reserve the rest for another use. Skim fat. Season with salt.
