In a small skillet, add the chopped apples, butter, and cinnamon. Cook over medium heat until the apples start to get tender - about 3 to 5 minutes. Do not overcook as they will continue to soften when baked. Remove the apples from the skillet and set aside (you can use the same skillet to make the brown butter).
In a small bowl, mix together the brown sugar and flour. Set aside.
Baking the Hand Pies - Preheat the oven to 425F degrees and adjust the oven rack to the middle position. Line a baking sheet with a piece of parchment paper.
Remove the cooled dough from the refrigerator and let it come to room temperature for at least 15 minutes. Lightly flour your work surface. Roll the dough into a rough 15-inch square. Using a 4.5-inch biscuit or cookie cutter, cut out 8 circles.
You will have extra dough. You can cut that into leaf shapes using a small cookie cutter to decorate the hand pies. You can re-roll the scraps as well to make more decorations, however, I do not recommend re-rolling the dough more than one time - otherwise, it will get tough.
Place 4 of the dough circles on the prepared baking sheet. Spoon equal amounts (about 2 and 1/4 teaspoons) of the brown sugar mix onto the center of each circle. Next, spoon equal amounts of the apple mix on top of the brown sugar - leaving at least a 1/4 inch border.
Place the remaining 4 dough circles on top and crimp the edges together with a fork. Lightly brush the tops of the pies with the beaten egg.
Bake the pies for 20 to 25 minutes, or until the tops are golden brown. Cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.
If you made pie cut-out decorations, bake them separately as they won't take as long to bake. Brush each with the beaten egg and sprinkle with cinnamon sugar for extra flavor. Bake for about 10 minutes, or until golden brown. Cool on a wire rack.