Prepare the Apple Filling: In a medium bowl, combine the diced apples, granulated sugar, brown sugar, cinnamon, and nutmeg. Toss everything together until the apples are evenly coated.

Preheat and Prep: Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper. This prevents sticking and makes cleanup a breeze.

Roll Out the Puff Pastry: Gently unfold one sheet of the thawed puff pastry on a lightly floured surface. Using a rolling pin, roll it out slightly to smooth out any creases and make it a little thinner.

Cut into Triangles: Using a pizza cutter or a sharp knife, cut the puff pastry sheet into 6-8 equal-sized triangles. You can do this by cutting it in half lengthwise, and then cutting each half into 3 or 4 triangles.

Brush, Spoon the Apple and Roll: Brush melted butter over. Spoon a small amount of the apple filling (about 1-2 tablespoons) onto the wide end of each triangle. Don't overfill, or the filling will spill out during baking. Starting at the wide end, roll up each triangle towards the point, forming a crescent shape.

Arrange and Egg Wash: Place the rolled croissants onto the prepared baking sheet, leaving a little space between each one. Brush the tops of the croissants with the beaten egg. This will give them a beautiful golden-brown color. If you like, sprinkle with coarse sugar.

Bake: Bake for 15-20 minutes, or until the croissants are puffed up and golden brown. Keep an eye on them towards the end of the baking time to make sure they don't over-brown.

Cool and Serve: Remove the croissants from the oven and let them cool on the baking sheet for a few minutes before transferring them to a wire rack to cool completely. They're best served warm, but they're also delicious at room temperature.