This is my favorite dish of all time, and one of the first I learned to make as a kid. My mother is the master of this dish and she taught me; her mother taught her, her mother’s mother taught her, and that’s how it works.
Bring the stock to a simmer in a large stock pot over high heat. Add the chicken, keeping the stock at a gentle simmer. Cover and cook for one hour or until the meat is falling from the bone. Remove the chicken from the pot, keeping the stock at a simmer. Remove the skin and meat (in big chunks) from the chicken. Discard the skin and the bones. Reserve the meat and set aside.
In a large mixing bowl, combine the flour and the buttermilk and season with black pepper. If the dough looks dry, add a few tablespoons of hot broth and a few additional tablespoons of buttermilk. Fold the dough lightly (it will look wet). Season the broth with salt and pepper as needed. Wet a spoon in the simmering broth, then scoop a heaping tablespoon of dumpling mixture and gently drop it into the broth over the largest simmering bubbles. Working quickly, repeat with the rest of the batter. Reduce to a low simmer, cover and cook for 10-15 minutes.
As the dumplings cook, baste as needed with broth. It's important that the dumplings don't overlap or sit on top of each other. Carefully push some of the dumplings to the side and add the chicken. Cover and simmer for another minute to heat through.
Check that the dumplings are cooked through by using a cake tester. Place dumplings, chicken and broth in a bowl. Finish with black pepper.