Put the chicken, onions, celery, carrots, garlic, and bay leaves in a large kettle. Cover with the water. Add the salt, black pepper, cayenne, and thyme and bring to a boil over high heat. Reduce the heat to medium and simmer for about 2 hours, or until the chicken is very tender. Remove from the heat.
Remove the chicken from the pot. With a slotted spoon, remove 1 cup of the vegetables from the pot. Set aside. When the chicken has cooled, skin and debone it. Set aside.
Beat together the egg and milk in a mixing bowl. Add the salt, pepper, baking powder, and parsley. Mix well. Add the flour and mix to make a thick batter. Fold the reserved vegetables into the batter. Set aside.
Remove the fat that has risen to the surface of the broth. Return the chicken meat to the pot and bring to a boil over medium-high heat. Dissolve the cornstarch in the milk and add to the pot. Stir and bring back to a boil. Reduce heat to medium.
Drop heaping tablespoons of the dumpling batter into the hot mixture, distributing the dumplings evenly, until all is used. When the dumplings rise to the surface, simmer and cook for 4 minutes, stirring gently, being careful not to break up the dumplings.
Remove the bay leaves and serve immediately in soup bowls.
Yield: 8 servings