In a large wide pot, simmer the chicken pieces in the broth until the chicken is tender, 30 minutes or more. Remove the chicken and set it aside.
When cool enough to handle, skin and bone the chicken and shred or cut it into bite sized pieces. If desired, skim the chicken fat off the surface of the broth.
In a large bowl, thoroughly mix the flour, baking powder and salt. Using a food processor, pastry blender, or two knives, cut in the butter until it is the size of very small peas and distributed throughout the dry ingredients. Add the water and knead 8-10 times. If the dough seems too sticky, add a little more flour.
Roll the dough out to 1/8-inch thickness and cut into 1-inch squares.
Bring the broth to a rolling boil, drop in the dumplings, cover the pot and reduce the heat to simmer. Cook for 10 minutes or until the dumplings are cooked. Overcooking will cause the dumplings to fall apart.
To serve, place the chicken pieces in a bowl and ladle hot broth and dumplings over the chicken.