FOR THE DOUGH:
Apple dumplings have roots in Pennsylvania Dutch country, where the pastries emerged as a clever use for leftover pie dough on baking day. They're so delicious, though, that we're happy to make up a fresh batch of pastry just for these alone. The sauce thickens as it cooks, turning into a rich and glossy syrup.
Cut two pieces of parchment paper so they're 11 inches wide and 16 inches long. Set aside.
Make the dough: In a medium bowl, whisk together flour, baking powder, salt, and sugar. Sprinkle the butter cubes and shortening on top and use your fingers to work them into the dough. Stop when the mixture resembles cornmeal with some pea-sized pieces of butter remaining. Add milk and stir, using a fork, until dough begins to hold together.
Turn the dough out onto one of the pieces of parchment paper; knead three times, or until the dough feels cohesive. Shape dough into a rough rectangle and cover with second piece of parchment paper. Roll dough out so that it is the same size as the parchment paper. Carefully transfer to refrigerator; let chill for 30 minutes.
FOR THE SAUCE:
Preheat oven to 425°;. Meanwhile, make the sauce: In a medium saucepan heat cider, rum, and brown sugar over high heat. Bring to a simmer and cook for 5 minutes. Remove from heat and set aside.
FOR THE APPLES:
Prepare the apples: In a small bowl, stir together brown sugar and cinnamon; set aside. Peel and core apples; trim if necessary to make each about 3 inches tall.
Remove the dough from the refrigerator and remove the top layer of parchment. Use a ruler and a knife to trim the dough to an exact 10-by-15-inch rectangle. Cut dough into six 5-inch squares. Set one apple in the center of each square; fill centers of apples with brown sugar mixture. Bring corners of dough up together around apples to make four points, gently sealing at top and along seams so that the pastry fits snugly.
Place each dumpling in a large baking dish. Pour sauce around dumplings and transfer to oven; bake for 10 minutes. Reduce oven temperature to 375° and bake until dumplings are golden brown and sauce is bubbling, 25 to 30 minutes more. Serve warm in bowls with extra sauce.