Cut each onion into 4 wedges, leaving roots and skin intact. Combine the onions and next 11 ingredients (onions through garlic) in a large Dutch oven or stockpot, and bring to a boil. Reduce heat, and simmer, uncovered, for 1 1/2 hours or until chicken is done and vegetables are tender. Strain the mixture through a cheesecloth-lined colander into a large bowl; discard the parsley sprigs and garlic.
Remove chicken from bones, and discard bones. Peel the onions. Place chicken-vegetable mixture in a bowl; cover and chill. Cover broth and chill at least 24 hours.
Skim the solidified fat from surface of broth, and discard. Prepare the Cornmeal-Herb Dumplings. Combine the broth, chicken, and cooked vegetables in Dutch oven, and cook over medium heat until thoroughly heated. Ladle soup into large shallow bowls; top with Cornmeal-Herb Dumplings. Garnish soup with fresh parsley and minced chives, if desired.