In a large pot over medium heat, combine chicken, celery and their tops, carrot, onion and its peel and pepper. Pour stock over and bring to a boil. Cover, reduce heat and simmer until chicken is tender and falls from the bone, about 45 minutes.
While chicken is cooking, make dumplings. In a large bowl, combine eggs, water, oil, salt and enough of the flour to make a stiff dough.
Strain chicken stock, reserving meat, celery and carrots. Pull meat from bones and return strained stock and meat, celery and carrots to pot. Bring to a boil.
Make dumplings by cutting dough from a stock-dipped spoon or using scissors or your fingers to make small, chickpea sized, dumplings and dropping them in the boiling water. When the dumplings rise to the surface they are done.