Place an oven rack in the center position and preheat your oven to 375°F (190°C). Grease a 9x9 baking dish with a little butter or non-stick spray.
Peel your two large apples, core them to remove the seeds and tough center, and cut each apple in half vertically so you end up with four halves. Set aside cut side up.
In a small mixing bowl, combine 1/4 cup brown sugar, 1 tsp cinnamon, and 1/4 tsp nutmeg. Stir until evenly blended. Scoop out about 1 tsp and set aside for later.
Unroll your refrigerated pie crusts onto a clean work surface. Cut each crust in half to create four equal portions, one for each apple half.
Place one apple half, cut side up, in the center of each pie crust portion. Spoon about 1 tablespoon of the brown sugar mixture into the cavity of each apple. Top each with one piece of butter. Fold the pie crust up and around the apple, pinching the edges together to seal completely. Place each wrapped apple, seam side down, in the prepared baking dish.
In a medium saucepan, combine 1 cup granulated sugar, 1 cup water, 1/2 tsp vanilla extract, and 1/4 tsp cinnamon. Bring to a gentle boil over medium heat, stirring occasionally. Reduce heat to low and simmer for 5 minutes until slightly thickened. Remove from heat and let cool slightly.
Carefully pour the warm cinnamon sauce over the dumplings in the baking dish, making sure each one is coated. Sprinkle the reserved 1 tsp of brown sugar mixture lightly over the tops. Bake in the preheated oven for 35-40 minutes until the crust is golden brown and the apples feel tender when pierced with a fork.
Remove the dish from the oven and let the dumplings cool right in the baking dish for about 10 minutes. Use a spoon to scoop any extra sauce from the dish over each dumpling before serving.