Place chicken in a large Dutch oven, and cover with water; bring to a boil. Cover, reduce heat, and simmer 2 hours. Uncover, and simmer an additional 30 minutes or until chicken is tender and broth is reduced by half. Remove chicken from broth; cool. Bone chicken, and cut into bite-size pieces. Set chicken and broth aside.
Combine next 3 ingredients; cut in butter with pastry blender until mixture resembles coarse meal. Add 1 cup milk; stir until well blended. Bring reserved broth to a boil. Drop batter by heaping tablespoonfuls into boiling broth, allowing room for dumplings to expand during cooking process. Cover immediately; reduce heat, and simmer 12 minutes or until a wooden pick inserted in center of each dumpling comes out clean. Repeat procedure with remaining batter, if necessary. Transfer chicken and dumplings to serving platter, and keep warm.
Cook remaining broth over high heat until reduced by half. Add 3 tablespoons flour to broth, stirring until smooth. Cook 3 minutes, stirring constantly. Gradually add 1 cup milk, stirring well. Cook over medium heat, stirring constantly, until thickened. Stir in 1/2 teaspoon salt and pepper. Pour gravy over chicken and dumplings.
Note: Dumplings are primarily cooked by steam heat, and should be kept covered during cooking. To prevent overcrowding of dumplings, allow room for expansion when placing batter in broth.