Combine the chicken pieces, carrots, onions, celery, parsley, pepper, salt, thyme springs, and bay leaves in a large Dutch oven. Pour the broth over the chicken mixture, making sure the mixture is covered completely. Bring to a boil over high heat. Cover, reduce the heat to low, and simmer until the chicken is done, about one hour and 15 minutes. Remove chicken. (Let the broth mixture continue to simmer). Let the chicken stand until cool enough to handle, about 10 minutes. Remove and discard the skin and bones from the chicken; shred the meat.

Whisk together the flour, baking powder, and salt in a large bowl. Add the milk and oil, stir until a stiff dough forms. Drop dough by teaspoonfuls into simmering broth mixture. Add the shredded chicken. cover and simmer until dumplings are cooked through but still tender, about 20 minutes. Discard bay leaves before serving.