Add the turkey carcass, water, broth, and quartered onion to a large stockpot and bring to a boil. Cover, reduce the heat, and allow the broth mixture to simmer for 1 hour.
Strain the broth into a large bowl, and discard the carcass and the onion. Cover the broth, and place into the refrigerator overnight. The following morning, skim the fat from the surface of the broth. Pour the broth into the stockpot.
Place the remaining ingredients into the stockpot and simmer, covered, for approximately 1 hour.
Using a ladle, scoop 2-cup portions of soup into individual bowls. Serve immediately.
