Prepare the apple filling: Preheat your oven to 375°F (190°C). Peel, core, and slice the apples into thin wedges. In a large bowl, toss the apple slices with lemon juice, granulated sugar, brown sugar, cinnamon, nutmeg, cornstarch, and a pinch of salt. Stir until the apples are evenly coated and the mixture thickens slightly.

1. Assemble the mini pies: Roll out the puff pastry sheets on a lightly floured surface. Cut each sheet into 4 squares (for a total of 8 squares). Spoon a generous amount of the apple filling into the center of each square, leaving about a 1-inch border. Place a few small cubes of butter on top of the apple filling.
2. Seal the pies: Fold the corners of each puff pastry square over the apple filling to form a rustic pocket or triangle. Pinch the edges to seal them. If desired, you can use a fork to press down the edges for a more decorative finish.
3. Egg wash and sprinkle: Beat the egg and brush the tops of the mini pies with the egg wash. Sprinkle with a little granulated sugar for extra sweetness and shine.
4. Bake the pies: Place the mini pies on a parchment-lined baking sheet and bake for 20-25 minutes, or until the puff pastry is golden and crispy, and the apple filling is bubbling.
5. Serve: Let the mini pies cool for a few minutes before serving. Enjoy them warm with a scoop of vanilla ice cream or a dusting of powdered sugar.