01 - Preheat oven to 400°F. Line a baking sheet with parchment paper.
02 - Melt butter in a small pan over medium heat. Add diced apple, brown sugar, cinnamon, and lemon juice. Cook, stirring frequently, for 3 to 4 minutes until the apples are soft and coated in a syrupy glaze. Remove from heat and allow to cool completely.
03 - In a mixing bowl, combine 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. Place squares on the prepared baking sheet. Using a knife, lightly score a border about 1/3 inch from the edge of each square without cutting all the way through.
05 - Spread approximately 1 tablespoon of cheese filling in the center of each pastry square. Top with a spoonful of the cooled apple mixture. Brush the pastry edges with beaten egg.
06 - Bake for 18 to 20 minutes until the pastries are puffed and golden brown.
07 - Allow danishes to cool for 10 minutes. Whisk powdered sugar with milk until a drizzle-consistency icing forms. Drizzle over the cooled danishes if desired.