01 - Preheat oven to 400°F. Line a baking sheet with parchment paper and set aside.
02 - Melt butter in a small pan over medium heat. Add the diced apple, brown sugar, cinnamon, and lemon juice. Cook, stirring frequently, 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. Stir until smooth and creamy with no lumps remaining.
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, lightly score a border about 3/8 inch from the edge of each square, 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, staying within the scored border. Spoon a portion of the cooled apple mixture on top of the cheese filling.
06 - Brush the exposed pastry edges generously with the beaten egg to promote a golden, glossy finish during baking.
07 - Bake for 18 to 20 minutes, or until the pastry is puffed and deeply golden brown. Rotate the baking sheet halfway through for even browning if needed.
08 - Allow the danishes to cool on the baking sheet for 10 minutes. In a small bowl, whisk together the powdered sugar and milk until a smooth, drizzle-able icing forms. Drizzle over the warm danishes and serve.