01 - Preheat oven to 400°F. Line a baking tray with parchment paper and set aside.
02 - In a skillet over medium heat, melt the butter. Add the diced apples, brown sugar, cinnamon, nutmeg, and lemon juice. Cook for 6 to 8 minutes, stirring frequently, until the apples are tender and the juices have reduced to a syrupy glaze. Remove from heat and let cool completely.
03 - In a mixing bowl, beat together the softened cream cheese, granulated sugar, and vanilla extract until smooth and creamy. Set aside.
04 - On a lightly floured surface, roll out the puff pastry sheet to smooth any creases. Cut into 8 equal squares. Place the squares on the prepared baking tray. Using a sharp knife, lightly score a smaller square border inside each pastry square without cutting all the way through. Prick the inner square several times with a fork to prevent excessive puffing.
05 - Spread a generous spoonful of the cream cheese filling in the center of each pastry square, staying within the scored border. Top the cream cheese with a spoonful of the cooled apple mixture.
06 - Brush the exposed pastry edges with the beaten egg to ensure a deep golden finish during baking.
07 - Bake for 18 to 22 minutes, or until the pastry is puffed and golden brown. Rotate the tray halfway through for even browning.
08 - Allow the danishes to cool on the tray for 10 minutes. In a small bowl, whisk the powdered sugar with enough milk or lemon juice to reach a drizzle consistency. Use a spoon to zigzag the icing over the cooled danishes.
09 - Serve warm or at room temperature. Danishes are best enjoyed fresh.