Boil chicken in 2 quarts of water, to which salt and butter have been added, until tender. Remove chicken, strain, and reserve broth. Remove skin of chicken and bone. Cut meat into large pieces.
Mix egg, salt, butter, and broth together. Add flour, mixing with a fork, a small amount at a time. Knead in flour until a stiff dough is formed. Roll out very thing. Cut in strips. Bring chicken broth to a boil. Pull apart and stretch strips of dough as you drop into broth. Reduce heat and cook about 10 minutes (covered), stirring occasionally. Add cut-up chicken to dumplings and enjoy!