Heat a 4 or 5 quart stockpot over medium-high heat. Season the chicken with salt and pepper; brown in some oil, about 3 minutes per side. Pour in the water; add the whole celery rib, onion, salt and pepper. Bring to a boil, then reduce to medium-low; cover and simmer until chicken is tender, about 1 hour. Remove chicken from broth. When cool enough to handle, remove chicken from bones, discard skin, and pull apart or cut into bite-size pieces. Strain the broth and skim off most of fat. Taste for seasoning.
Bring the broth to a boil in the same pot. Add the chopped onions and celery. Reduce heat and simmer, covered, until tender, about 25 minutes. Add the reserved chicken. If desired, add flour and water paste to thicken broth slightly but not too much as the dough will also thicken it.
Add choice of Dumplings:
Light and Fluffy Dumplings
2 cups sifted flour
1/2 teaspoon salt
4 teaspoons baking powder
2 teaspoons dried parsley, crushed with hands
Pepper to taste
3 tablespoons butter
1 large egg, beaten
1/2-2/3 cup milk
Whisk together sifted flour, salt and baking powder. Add the parsley and pepper. Cut in butter with two knives or pastry cutter. Stir in egg. Gradually add enough milk to make a moist but fairly stiff dough. Drop dough by tablespoons into gently simmering chicken mixture. Cover tightly and simmer for 12-15 minutes, or until dumplings are done. Serve immediately.
Heavy Dumplings
Source: Fannie Farmer Cookbook
2 cups flour
3 tsp baking powder
1 tsp salt
2 tsp. minced parsley
4 T shortening
3/4 to 1 cup milk
Combine dry ingredients and parsley.
Cut in shortening until it looks like coarse meal.
Stir in 3/4 cup milk with a fork - add remaining milk if the dough will not hold together.
Drop dumplings by spoonful into pot. Cover and simmer for 20 minutes without lifting lid.