Combine flour, salt, soda, and sugar; stir well. Cut in shortening with a pastry blender until mixture resembles coarse meal. Stir in buttermilk and egg. Shape mixture into a ball; cover and place in freezer for 30 to 40 minutes.
Combine chicken, broth, water, and pepper in a large stock pot. Bring to a boil. Reduce heat; cover and simmer 35 minutes or until chicken is tender. Remove chicken from broth; cool. Bone and skin chicken; cut meat into bite-size pieces, and set aside. Strain broth, and return to stock pot.
Turn dumpling dough out onto a heavily floured surface. Roll dough to 1/8-inch thickness; cut into 1-inch squares. Bring chicken broth to a boil. Drop dumpling squares into broth. Add chicken pieces. Reduce heat; cover and simmer 20 minutes. (Do not remove lid or dumplings will be tough.) Stir in milk. Serve immediately.