01 - Preheat oven to 400°F. Line a baking tray with parchment paper and set aside.
02 - Melt butter in a skillet over medium heat. Add diced apples, brown sugar, cinnamon, nutmeg, and lemon juice. Cook for 6 to 8 minutes, stirring frequently, until apples are tender and coated in a thick syrup. Remove from heat and let cool completely.
03 - In a mixing bowl, beat softened cream cheese, granulated sugar, and vanilla extract together until smooth and creamy. Set aside.
04 - Unfold puff pastry onto a lightly floured surface and roll gently to smooth out creases. Cut into 8 equal squares. Transfer squares to the prepared baking tray. Using a knife, lightly score a smaller square border inside each piece without cutting all the way through. Prick the inner square several times with a fork to prevent puffing.
05 - Spread a generous spoonful of cream cheese filling in the center of each pastry square, staying within the scored border. Top with a spoonful of the cooled apple mixture.
06 - Brush the exposed pastry edges with beaten egg to promote even browning during baking.
07 - Bake for 18 to 22 minutes until pastry is puffed and deeply golden brown. Transfer to a wire rack to cool slightly.
08 - Whisk powdered sugar with enough milk or lemon juice to reach a drizzle consistency. Spoon zigzag lines over the cooled danishes. Serve warm or at room temperature.