Heat the chicken broth over high heat in a large stockpot with a lid. Remove the skin from the chicken and pull the meat off the bone. Shred the chicken meat.
Add the chicken to the broth and bring to a boil.
Meanwhile, combine the flour, sage, thyme, oregano, and pepper to the bowl of a food processor fitted with the metal blade. Pulse 1 time. Add the shortening and pulse 6 times.
Add the milk and process until the dough comes together, about 30 seconds.
Remove the dough from the food processor and place on a lightly floured countertop.
Use a floured rolling pin to roll the dough to ¹⁄8-inch thickness. Use a pizza cutter to cut into strips about 1 by 3-inches. Drop the strips into the boiling broth a few at a time. The dumplings will look waterlogged and gooey.
Stir very gently just once. Reduce the heat to medium.