1. Cut two pieces of parchment paper to 16 by 11 inches. Set aside.

2. Make the dough: In a medium bowl, whisk together flour, baking powder, salt, and sugar; sprinkle butter cubes and shortening over top of flour mixture. Using your fingers, work butter and shortening into the flour, smearing as you go, until mixture resembles cornmeal with some pea-size pieces of butter remaining. Add milk and stir, using a fork, until dough begins to hold together.

3. Turn dough out onto one of the pieces of parchment paper; knead until dough feels cohesive, about 3 times. Shape dough into a rectangle and cover with second piece of parchment paper. Roll dough between parchment paper into a 16-by-11-inch rectangle. Carefully transfer to refrigerator; let chill for 30 minutes.

4. Preheat oven to 425 degrees.

5. Make the sauce: Meanwhile, place a medium saucepan over high heat. Add cider, rum, and brown sugar. Let simmer for 5 minutes. Remove from heat and set aside.

6. Make 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.

7. Using a ruler and a sharp knife, trim chilled dough into a 10-by-15-inch rectangle. Cut 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.

8. 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 degrees and bake until dumplings are golden brown and sauce is bubbling, 25 to 30 minutes more. Serve warm in bowls with extra sauce.