Heat the oil in a large, heavy-bottomed Dutch oven over medium-high heat. Combine the flour, salt, and pepper in a medium bowl. Dredge the chicken in flour mixture, and turn to coat; shake off excess flour. Add the chicken to the pot; cook 3 minutes on each side, or until well-browned. Transfer chicken to a plate.
Add leeks to pot; cook over medium heat 2 minutes, stirring frequently. Return chicken to pot, and stir in broth, carrots, parsnips, celery, thyme, and bay leaf. Bring to a boil. Partially cover; reduce heat. Simmer 15 minutes, or until chicken is cooked. Remove chicken from pan. Pull the meat from bones; discard bones. Return the chicken to the pot, and stir in peas.
To prepare the dumplings, sift together the flour, baking powder, and salt 3 times (this makes the dumplings very light). Combine the flour mixture, milk, and chives, and stir until moist.
Drop dough by 3 tablespoonfuls into a barely simmering chicken mixture. Cover and cook over medium-low heat 12 minutes, or until the dumplings are done (do not bring to a boil, or the dumplings will break up).