Prep the Oven and Pan: Preheat your oven to 375°F. Generously butter a 9x13 inch baking dish and set it aside.
Prep the Dough: Roll out the pie crusts and divide them into 6 equal pieces (roughly 7-inch squares or circles). Ensure the dough pieces are large enough to completely encase your apples. Place one peeled and cored apple in the center of each dough piece.
Mix the Spices: In a small bowl, whisk together the ½ cup brown sugar, 1 teaspoon cinnamon, ½ teaspoon nutmeg, and ½ teaspoon salt.
Stuff the Apples: Evenly divide the brown sugar spice mixture among the six apples, packing it firmly down into the cored centers and sprinkling any extra around the base of the apples. Place one slice (1 tablespoon) of butter directly on top of the opening of each stuffed apple.
Wrap the Dumplings: Carefully fold the edges of the pie dough up and over the apple. Pinch the seams tightly to seal the apple completely inside the dough. Transfer the finished dumplings to the prepared baking dish, spacing them evenly.
Make the Sauce: In a medium saucepan, combine the apple cider, 3 tablespoons brown sugar, 2 tablespoons butter, vanilla extract, and 1 teaspoon cinnamon. Heat over medium-high heat, stirring constantly, until it reaches a low boil and the sugar is completely dissolved. Boil for 1 minute, then remove from the heat. Let it cool for 5 minutes.
Glaze and Bake: Carefully pour the slightly cooled cider sauce evenly over the top of the dumplings in the baking dish. Place the dish in the preheated oven and bake for 50 to 55 minutes, or until the pastry is deeply golden brown and the apples are tender when pierced with a skewer.
Rest and Serve: Remove from the oven and allow the dumplings to rest in the syrup for 10 minutes before serving. Serve warm, topped with the pan syrup and a scoop of vanilla ice cream.