01 - Preheat the oven to 400°F. Line a baking sheet with parchment paper and set aside.
02 - In a small pan, melt the butter over medium heat. Add the diced apple, brown sugar, cinnamon, and lemon juice. Cook while stirring for 3 to 4 minutes until the apples are tender and coated in a syrupy glaze. Remove from heat and allow to cool completely.
03 - In a mixing bowl, combine the softened cream cheese, granulated sugar, and vanilla extract. Beat until smooth and creamy.
04 - Roll out the puff pastry sheet on a lightly floured surface and cut into 8 equal squares. Transfer the squares to the prepared baking sheet. Using a sharp knife, score a border approximately 3/8 inch from each edge, being careful not to cut all the way through the dough.
05 - Spread approximately 1 tablespoon of the cheese filling in the center of each pastry square. Top with a generous spoonful of the cooled apple mixture, keeping the filling within the scored border.
06 - Brush the exposed pastry edges with the beaten egg to ensure a golden, glossy finish after baking.
07 - Bake for 18 to 20 minutes until the pastry is deeply golden and fully puffed. Transfer to a wire rack and cool for 10 minutes.
08 - Whisk together the powdered sugar and milk until a smooth, drizzle-able icing forms. Adjust consistency with additional milk if needed. Drizzle over the cooled danishes and serve warm or at room temperature.