Place flour in a large resealable plastic bag; add chicken pieces, a few at a time and shake to coat. In a large skillet, brown chicken in batches in oil; drain.
Place chicken in an 8-qt. stockpot. Add the celery, carrots, parsley and seasonings. Add enough water to cover chicken; bring to a boil. Reduce heat; cover and simmer chicken for 45-50 minutes or until no longer pink.
Remove 1 cup broth to use for dumplings; cool. Stir in the flour, baking powder and eggs to form a stiff batter; drop by tablespoonfuls onto simmering broth. Cover and simmer for 15-20 minutes or until a toothpick inserted in a dumpling comes out clean (do not lift the cover while simmering). Remove chicken and dumplings to a serving dish and keep warm.
For gravy, remove 4 cups broth and vegetables to a large saucepan; bring to a boil. Combine flour and water until smooth. Gradually stir into vegetable mixture. Bring to a boil. Cook and stir over medium heat for 2 minutes or until thickened and bubbly. Serve with chicken and dumplings. Yield: 6-8 servings.