Place chicken in a large pan; add water, salt and pepper. Bring to a boil; reduce heat and simmer until tender and juices run clear when chicken is pierced with a fork, about one hour.
Remove chicken, reserving broth in a pan. Let chicken cool; remove bones and return meat to chicken broth. Add chopped eggs. Bring broth to a boil and add Dumplings one at a time; stir well before adding each new batch of dumplings.
After adding last batch, cover and simmer until tender, about 20 minutes. Remove from heat; let stand a few minutes before serving.
Dumplings: Mix flour with enough water to make a dough that can be rolled out. Divide dough into 4 batches. Roll out each batch of dough 1/2-inch thick on a lightly floured surface; cut into strips.