To make the stock and filling, discard any excess fat from the chicken cavity. Rinse the chicken well, then place in a stockpot with the water. Add the celery, onion, carrot, and herbs to the pot, bring it to a boil, and immediately reduce the heat to a simmer. Cook, skimming the pot frequently, until the meat is thoroughly cooked, about 1 hour.
Remove the chicken and the celery ribs fr