Stew one whole chicken in a large pot of water. Reserve bag of giblets for gravy. (This assumes that you are making dumplings for a large holiday or family gathering and would need gravy. Anyway, save for another use.) Season pot with salt and pepper and stew at a strong simmer until meat is falling off the bone. Strain meat and bones out of broth and pick off meat. Set aside.
1 cup rich broth
all purpose flour
Add enough flour the the broth and egg to make a soft dough. Take care not to work it any more than necessary to form dough (to avoid tough dumplings). Roll out thin and cut into 3" x 1" strips.
Bring broth to a rolling boil.. Gently drop in dumplings a few at a time to maintain a low boil. Don't stir, but carefully press dumplings down a few times to cook them evenly. Keep checking dumplings until they are done. This could be from 10-30 minutes depending on how tough your dumplings were to start with. Add as much meat back in as you like.
Pictures are of Grandma giving Mark a dumpling-making lesson and then eating them with Valerie. The date on the pictures is wrong because Val brought McKenna with them.