Place chicken, water, bouillon, celery, onion and pepper together in pot.
Cook, covered for 1 hour. (After cooking, I let contents remain in covered pot for about an hour to cool).
As soon as chicken is done cooking, remove 1 cup of cooking liquid from pan and allow to cool so that it is about room temperature.
In a bowl mix self-rising flour and the 1 cup cooled broth.
You may need to add a little more flour if mixture is too sticky.
On a floured surface, roll out dough to 1/4 inch thickness (I divide dough into 1/3's to roll out, it's easier to do smaller amount) Cut rolled dough into 1 inch strips, then cut strips into 3 inch pieces.
Place cut strips on a cookie sheet and place in freezer for 10-20 minutes.
While dumplings are chilling, remove chicken from pot.
Heat broth in pot to a boil.
Remove dumplings from freezer and drop into boiling broth, after all the dumplings are in pot lower heat so that they are just simmering.
Cover pot, and cook for twenty minutes.
Do not uncover to peek!
While dumplings are cooking, cut chicken into bite size pieces.
After dumplings have cooked for 20 minutes, add chicken back to pot.