How to Make It
Preheat the oven to 375°. Line a large rimmed baking sheet with parchment paper. Cut each apple quarter into 5 lengthwise slices.
On a lightly floured work surface, roll out the puff pastry 1/8 inch thick, working in batches if needed. Using a bowl or plate as a template, cut out six 6 1/2-inch rounds. Put the rounds, slightly overlapping, on a cookie sheet and refrigerate until firm, about 10 minutes.
Arrange about 6 of the apple pieces in the center of each round in a loosely packed pile, leaving a 1 1/2-inch border. Tuck 2 small apple pieces in the center. Fold the pastry up and over the apples to partially cover them. Transfer the tartlets to the lined baking sheet and brush the pastry with the egg wash. Bake for about 45 minutes, or until the apples are tender and the pastry is puffy and browned. Sprinkle the tartlets with the brown sugar and let cool on the sheet for 5 to 10 minutes. Serve hot or warm.