1. Preheat the oven to 400°F (200°C) and line a baking sheet with parchment paper.

2. In a large skillet, combine the diced apples, lemon juice, 3 tablespoons granulated sugar, brown sugar, cinnamon, nutmeg, 1/8 teaspoon salt, and cornstarch.

3. Cook the apple mixture over medium heat, stirring often, until the apples are just tender and the juices have thickened and look glossy, about 6 to 8 minutes.

4. Transfer the apple filling to a shallow bowl or plate and let it cool completely to room temperature.

5. In a mixing bowl, beat the softened cream cheese with 1/3 cup granulated sugar until completely smooth and creamy.

6. Add the vanilla extract, milk or cream, and a small pinch of salt if desired, then beat again until the cheesecake filling is thick but easily spreadable.

7. Lightly flour your work surface and unfold the thawed puff pastry sheet.

8. Roll the pastry gently with a rolling pin to smooth the seams and slightly enlarge the sheet.

9. Cut the pastry into 6 equal rectangles, then place 3 of the rectangles onto the prepared baking sheet to act as the bottoms of the pies.

10. Spoon a generous tablespoon of cheesecake filling into the center of each pastry rectangle, leaving a clean border around the edges.

11. Top the cheesecake layer with a spoonful of cooled apple filling, mounding it neatly in the center but keeping it away from the edges.

12. In a small bowl, whisk together the beaten egg and 1 tablespoon water to make an egg wash.

13. Brush the exposed pastry edges around each filling mound with a thin layer of egg wash to help seal the pies.

14. Place the remaining pastry rectangles over the filling on each bottom piece, gently stretching the dough if needed to cover the apples.

15. Press the edges firmly together with your fingers, then crimp all the way around each pie with the tines of a fork to seal completely.

16. Use a sharp knife to cut two or three small slits in the top of each pie to allow steam to escape during baking.

17. Brush the tops of the pies with more egg wash and sprinkle with coarse sugar if you like extra crunch and shine.

18. Bake the Cheesecake Stuffed Apple Pies for 18 to 22 minutes, or until the pastry is puffed, deep golden brown, and the filling is just starting to bubble at the vents.

19. Remove the baking sheet from the oven and let the pies cool for 5 to 10 minutes.

20. Transfer the pies to a wire rack and dust with powdered sugar, or whisk the powdered sugar with a splash of milk and vanilla to create a quick glaze to drizzle over the tops.