Combine first 3 ingredients with 2 teaspoons salt, and 1/4 teaspoon pepper in a large Dutch oven. Bring to a boil. Cover, reduce heat, and simmer 2 1/2 hours or until chicken is tender. Remove chicken from broth; cool. Bone chicken, and cut into bite-size pieces; set aside. Strain broth.
Combine 1/2 cup flour and 1 cup water, blending well; add to broth in Dutch oven, stirring until smooth. Stir in chicken.
Combine remaining flour, salt, and pepper; stir well. Cut in shortening with a pastry blender until mixture resembles coarse meal. Stir in egg and milk. Bring broth mixture to a boil. Drop dumpling batter by tablespoonfuls into broth mixture, allowing room for expansion. Cover; reduce heat and simmer 15 minutes or until a wooden pick inserted in center comes out clean. Repeat procedure with remaining batter.