Boil chicken in a pot, adding salt to taste. When tender, pull chicken from the bone and return to the pot, adding broth.
In a bowl, combine shortening and flour, mix together with a pastry cutter, then add buttermilk to form a doughy consistency (similar to how you would make biscuits). Sprinkle flour on cutting board. Pinch off part of the dough and roll into a thin sheet and cut into squares. Drop one square at a time into broth. Continue this until all of the dough is used. Add salt and pepper to taste and simmer until tender.
Allow the dumplings to thicken the broth for 2 hours until it is a creamy consistency.