Place chicken in a Dutch oven with onion, carrots, celery, garlic, bay leaves, thyme, parsley, 1 1/2 teaspoons salt, black pepper, and cayenne pepper. Add enough cold water to cover. Bring just to a boil over medium-high heat. Reduce the heat to low, cover, and simmer until tender, about 1 hour.
Remove chicken, and let rest until cool enough to handle. Skin and bone chicken, leaving the meat in large chunks; set aside.
Remove bay leaves from the broth, and discard. Skim as much fat as possible from the surface of the broth. Heat broth to a slow steady boil. In a medium bowl, combine flour, baking powder, and remaining 1/2 teaspoon salt. Stir in milk, and beat until stiff. Drop batter 1 tablespoon at a time into the boiling broth. Cover, and cook for 10 minutes. Add reserved chicken, and cook until heated through, about 5 minutes. The dumplings should be puffed and the meat warmed through. Garnish with parsley, and serve immediately.
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