Line a baking sheet with parchment paper and set it aside. Take your thawed puff pastry out of the refrigerator. It should feel cold and firm, not warm or sticky. Unfold it onto a lightly floured surface and use a rolling pin to gently roll it into a rectangle about 10 by 12 inches.

In a small bowl, stir together the softened butter, granulated sugar, brown sugar, cinnamon, and a pinch of salt until it forms a paste. Spread this mixture evenly over the entire surface of the pastry, going all the way to the edges.

Scatter the finely chopped apple pieces evenly over the cinnamon butter layer. Press them down gently with your palm so they stick to the butter and do not roll around when you start rolling the dough.

Starting from one long edge, roll the pastry into a tight log. Pinch the seam firmly to seal it. Wrap the log in plastic wrap and place it in the freezer for 15 minutes. This step is important. Cold dough slices cleanly without squashing the spiral.

Preheat your oven to 400 degrees while the log chills. After 15 minutes, remove the log from the freezer and unwrap it. Use a sharp knife to slice the log into rounds about 3/4-inch thick. You should get about 12 slices.

Place the rounds cut side up on the prepared baking sheet, spacing them about 1 1/2 inches apart. Brush the tops lightly with the beaten egg. Sprinkle with coarse sugar if using.

Bake for 18 to 20 minutes until the pinwheels are deeply golden brown on top and the layers look visibly flaky. The bottoms should be golden as well. Let them cool on the pan for 5 minutes before serving. They firm up as they cool.