One of the many cherished recipes my mama taught me, I have tweaked it a few times. If you are frightened of rolling dough or don't have the "feel" for making it, try Already Pie Crust. Also, substitute cream or Half and Half for the milk. This along with real Butter makes for very rich Dumplings.
You can do this ahead of time and it makes for faster prep time; Cut up fryer into four pieces, cover with water and boil until done. Let cool, remove chicken from water/broth, skim top of broth removing oils. Remove skin from chicken and de-bone tearing or cutting into bite sized pieces. Place in refrigerator, set broth aside.
In medium sized bowl, add flour and butter/margarine. Cut butter into flour with pastry cutter or fork until it resembles fine cornmeal. Add egg, mix thoroughly with flour mixture until the crumbs are a bit larger. Add milk, SLOWLY, mixing with fork until it starts to pull away from sides of bowl and clings together into a ball. If it sticks to your hands, work in a few tablespoons of flour. If it seems dry, add a little more milk. You can also use cream if you want the dumplings rich. Once you have your pastry in a ball, roll out on a piece of waxed, lightly floured paper until it is less than half an inch thick. About the thickness of a pie crust. You don't have to be exact on a perfect square or round pastry. Cut pastry into 2" strips. This makes quite a lot of dumplings. And if you've gotten it right you can put one on top of another and they won't stick. This just makes it easier to carry to stove on a plate.
Add canned broth to boiled chicken broth. Add chicken, bring mixture to a boil on medium heat. Add Dumplings, one at a time. They should hold together at this point. Make sure they do before you add any more. If you use a pie crust, no worries at all. But, be brave and make your own pastry! Continue to boil on Medium Heat until Dumplings are done. To test, remove one, cut it in half. If it is not gummy, it's done. This usually takes about thirty minutes. Turn off Heat. Now, if you want to, add a bit of Half and Half or Milk. I don't do this but there are some that do. Too rich for me. These are excellent. Serve warm. Refrigerate leftovers.