Remove chicken innards and place in a cheesecloth bag. Tie shut.
Fill a large saucepan or stock pot with water. Add onions, chicken, and bag of innards, and salt. Bring to a simmer for 1½ to 2 hours.
Remove chicken and bag of innards from pot. Discard innards. Carve meat off the chicken (discarding any excess fat), and add back to pot. Add vegetables one half-hour before serving.