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

2. In a skillet over medium heat, melt the butter, then add the diced apples, brown sugar, cinnamon, and lemon juice. Cook for 5 to 7 minutes until the apples are tender. Let the filling cool.

3. In a mixing bowl, beat the cream cheese, granulated sugar, vanilla extract, and egg yolk until smooth.

4. Unfold the puff pastry on a lightly floured surface and cut it into 6 equal rectangles. Transfer them to the prepared baking sheet.

5. Lightly score a smaller border inside each rectangle, being careful not to cut all the way through.

6. Spoon the cream cheese mixture into the center of each pastry, then top with the cooled apple filling.

7. Whisk together the whole egg and water, then brush the edges of the pastry with the egg wash.

8. Bake for 16 to 20 minutes, until the pastries are puffed and golden brown.

9. Let the danishes cool slightly.

10. Stir together the powdered sugar and milk until smooth, then drizzle over the warm danishes before serving.