Heat the oven to 375°F. Lightly butter a 9×13-inch baking dish.
Prepare the apples. Peel and core the apples, keeping each apple whole.
In a small bowl, mix the brown sugar, cinnamon, nutmeg, and salt.
Divide the pie dough into 6 pieces large enough to completely wrap an apple. Roll each piece into roughly a 7-inch square.
Place an apple in the center of each piece of dough. Put about 1/2 tablespoon butter into the center of the apple and generously spoon in the cinnamon-sugar mixture.
Bring the corners of the dough up over the apple. Pinch the seams together so the apple is completely enclosed. Place the dumplings seam-side up in the baking dish.
Make the Sauce
Combine the water, granulated sugar, butter, and cinnamon in a small saucepan.
Bring to a gentle boil and cook for about 2–3 minutes, stirring until the sugar dissolves.
Remove from the heat and stir in the vanilla.
Pour the warm sauce around the dumplings, rather than directly over the tops, so the pastry stays attractive and crisp.
Bake
Bake uncovered for 40–45 minutes, until the pastry is deeply golden and the apples are tender when pierced with a knife.
Let them rest for about 10 minutes before serving.