Heat oil in a large stockpot over medium heat.

Add carrots, onions, leeks and celery and cook until just softened, about 5 minutes.

Add chicken and continue to cook, turning occasionally, until just golden brown, about 10 minutes more.

Add 2 cups water, broth, pepper flakes (if using), rosemary and thyme and bring to a boil.

Reduce heat to medium-low, cover and simmer for 2 hours.

Transfer chicken to a large plate and set aside to let cool.

Meanwhile, remove and discard rosemary and thyme from broth and skim off fat from the surface, if needed.

When cool enough to handle, remove chicken meat from bones and shred into small pieces.

Discard bones and return chicken to pot with broth; bring to a boil.

Meanwhile, combine flours, cornmeal, baking powder, salt and pepper into a medium bowl and mix well.

Add butter or shortening and cut into flour mixture with a pastry cutter or your fingers until mixture resembles course meal.

Stir in chives, then add egg and milk, stirring slowly just until a soft dough forms. (Be careful not to overmix or the dumplings will be tough.)

Carefully drop walnut-size spoonfuls of the dough into the hot stock, then cover the pot tightly and gently simmer until dumplings are puffed and cooked through, about 20 minutes.

Ladle chicken and dumplings into deep bowls and garnish with parsley.