Add the chicken pieces, celery, carrots, onion, salt, and pepper in a pot and add enough water to cover chicken. Bring to a boil then reduce heat to a simmer for 1 hour or until the chicken is cooked through. There, you just made chicken broth and cooked the chicken.
Remove the pot from heat and let cool, THEN remove the chicken with a slotted spoon to a plate and remove the skin and bones (makes for juicy chicken), skin and bones, great treat for your dog. Return the meat to the pot, taste for salt and adjust as needed. Return the soup to a simmer.
Mix the dumpling ingredients together and mix to form a stiff ball, drop by tablespoons into simmering soup, cover and simmer for 15 to 20 minutes.
Serve with bread and a salad for a complete meal.