Add chicken, salt and pepper to large stock pot. Add enough water to cover chicken completely. Bring to a boil, then reduce heat to medium-low. Cover, and simmer for about 45-60 minutes, or until chicken is tender and cooked through. If desired, remove chicken from broth until it's cool enough to handle, then shred and return to pot.
Add butter to broth and cover. If broth is not boiling, increase heat to medium-high.
In a medium bowl, sift flour to remove lumps. Slowly add milk to make a stiff dough. Flour both hands, and roll about a heaping tablespoon of dough into a ball, then press gently with fingers to flatten. Drop dumplings, one at a time, in boiling broth. Cover; reduce heat to low. Simmer 15 minutes or until dumplings are cooked through. Adjust salt and pepper as needed.