Bring chicken, water, celery, carrots, onion, parsley, rosemary, garlic, and butter to a boil, reduce heat and simmer uncovered for at least 3 hours or until the chicken falls off the bones. Make sure to skim the foam as needed.
Meanwhile, in a large mixing bowl, combine flour and salt. Cut shortening into flour until fully incorporated. Add buttermilk slowly into the flour a little at the time. When the dough is easily formed into a ball, knead the dough.
Roll out the dough to a thickness of 1/16 of an inch or even thinner. Pinch or cut into 1 inch squares (I use a pizza cutter for this). Set aside.
When chicken is falling off the bone, remove the pieces to a plate. Strain the broth throwing away vegetables and herbs.
Add broth back to the pot and add butter, salt and pepper. Bring to a boil and lower to a simmer. Pull all the meat from the chicken and shred. Return chicken to the broth.
Add the dumplings to the broth and bring to a hard boil. Allow them to boil about 2 minutes and turn off the heat. Stir gently with a wooden spoon every few minutes for the next 30 minutes. This step releases the starch in the dumplings, promotes creaminess, and intensifies the buttery flavors!