Place hen or chicken in a large pot with enough water to cover. (You could also use several chicken breasts instead of whole chicken.).
Cook on low-medium heat until done. Remove the chicken from the pot. Keep the broth in the pot. You will cook the dumplings in this broth.
Cool the chicken until you can remove the skin and bones. (I usually shred the chicken, but you can cube it if you wish.).
Meanwhile, mix the flour, salt and shortening with a fork until it resembles crumbs.
Add milk and mix well.
Roll out to about 1/8 inch on floured waxed paper or on a floured counter.
Let set for about 30 minutes while you debone and shred cooked chicken.
Cut the dough into strips. (I usually use a pizza cutter for this & cut the strips about 1"x2 or 3" wide.).
Put the chicken and 1/3 cup butter into the broth in the pot.
Bring the broth to a boil.
Drop the dough strips into the boiling broth, one at a time.
Simmer for 30 minutes, then serve in bowls.
NOTE: I usually double the amount of dumpling dough. My family likes lots of dumplings. I have made this ahead by freezing the cut dumplings on a cookie sheet, then transferring to a zip-lock bag. Freeze the broth with shredded chicken in a separate container. When you're ready to eat them, just heat the broth & chicken to a boil (thaw dumplings while heating broth) and drop in the dumplings & cook according to directions.