Combine first 7 ingredients in a Dutch oven, and cover with water; bring to a boil. Cover; reduce heat, and simmer 4 hours or until chicken is tender. Remove chicken from broth; cool. Bone chicken, and cut into bite size pieces. Set aside. Strain broth, and set aside.
Combine 1/4 cup plus 1 tablespoon flour and 2 tablespoons melted butter; stir well to make a paste. Gradually add paste to broth, stirring until smooth. Cook over medium heat, stirring constantly, until thickened and bubbly. Stir in chicken.
Combine 1 cup flour, baking powder, baking soda, 1 tablespoon butter, and milk; stir well. Drop batter by tablespoonfuls into boiling broth mixture, allowing room for dumplings to expand during cooking process. Cover immediately; reduce heat, and simmer 20 minutes or until a wooden pick inserted in center of each dumpling comes out clean. Repeat procedure with remaining batter, if necessary. Serve immediately.