1. In a large stockpot, heat olive oil or butter. Add carrots, onions, leeks, and celery. Saute for 5 minutes over medium heat. Add chicken quarters. Continue to cook, browning the chicken, for another 5 to 10 minutes. Add water, broth, chili flakes, rosemary, and thyme. Bring to a boil, then reduce heat to simmer, covered, for 2 hours.
2. Remove chicken to a large bowl or plate and set it aside to cool. Leave broth on the stove, but with no heat. Remove rosemary stems and thyme stems and discard. If desired, skim some of the fat off the surface of the broth.
3. Remove chicken meat from bones, shredding or chopping into bite-size pieces. Discard bones. Return chicken to stockpot. Bring to a boil.
4. Prepare dumplings by combining the flours, cornmeal, baking powder, salt, and pepper in a medium bowl. Cut in the butter or the shortening. Stir in the chives. Add the egg, and then slowly add the milk until a soft dough forms. Do not over-mix the dough.
5. Carefully drop walnut size dumplings into stock. Cover tightly and simmer 20 minutes. These will expand and rise to the top of the liquid.
6. Using a slotted spoon, transfer chicken and dumplings to a large platter. Add hot stock from the pot, if desired. (Leftover stock makes a delicious soup!)
7. Garnish with parsley and serve.