Combine chicken, 4 cups water, onion, celery and leaves, carrot, salt, and pepper in large kettle or Dutch oven with tight-fitting cover.
Cover; heat to boiling, then simmer 1 1/2 to 2 hours, or until chicken is tender.
Remove from broth; cool slightly; slip off skin, if you wish.
Strain and measure broth; add water, if needed, to make 5 cups.
Press vegetables through strainer into broth in kettle; heat to boiling.
Stir 1/2 cup cold water into flour in cup to make a smooth paste; stir into hot broth.
Cook, stirring constantly, until gravy thickens and boils 1 minute.
Season with salt and pepper to taste, if needed.
Return chicken to gravy in kettle; heat slowly to boiling while stirring up Dappled Dumplings.
Drop dough in 12 mounds on top of steaming chicken.
Cook, covered, 20 minutes.
(No peeking, or the dumplings won't puff properly.) 6.
Arrange chicken and dumplings on a heated serving platter; pass gravy in separate bowl.
DAPPLED DUMPLINGS: Sift 2 cups sifted flour, 3 teaspoons baking powder, and 1 teaspoon salt into medium-size bowl.
Cut in 2 tablespoons shortening with pastry blender until mixture is crumbly.
Stir in 1/4 cup chopped parsley and 1 cup milk just until flour is moistened.
(Dough will be soft.)