01 - Preheat oven to 400°F. Line a baking sheet with parchment paper.
02 - Melt butter in a small pan over medium heat. Add the diced apple, brown sugar, cinnamon, and lemon juice. Stir continuously for 3 to 4 minutes until the apples are tender and coated in a syrupy glaze. Remove from heat and let 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 the edge of each square without cutting all the way through.
05 - Spread 1 tablespoon of the cheese filling in the center of each pastry square. Top with a generous spoonful of the cooled apple mixture.
06 - Brush the exposed pastry edges with the beaten egg to ensure a golden, glossy finish.
07 - Bake for 18 to 20 minutes until the pastries are puffed and golden brown.
08 - Allow the danishes to cool for 10 minutes. Whisk together the powdered sugar and milk until a smooth icing forms. Drizzle over the warm danishes as desired.