Thaw the puff pastry until flexible but still cool. Heat the oven to 400°F (200°C), and line a baking sheet with parchment paper.

Melt the butter in a skillet over medium heat. Add the diced apples, brown sugar, cinnamon, lemon juice, flour or cornstarch, and salt. Cook for 5–8 minutes, stirring often, until the apples are slightly softened and the juices have thickened. Cool the mixture completely before assembly.

Beat the softened cream cheese, granulated sugar, egg yolk, and vanilla extract until smooth and creamy.

Cut each puff pastry sheet into four equal squares, creating eight squares total. Place them on the prepared baking sheet and lightly score a smaller square inside each piece without cutting all the way through.

Place about 1 tablespoon of cream-cheese mixture in the center of each pastry square, followed by a spoonful of cooled apples. Do not overfill, since excess filling can spill onto the baking sheet.

Beat the whole egg with 1 tablespoon of water. Brush the exposed pastry edges with the egg wash, taking care not to let it run down the cut sides.

If the pastry has become soft or warm, chill the assembled danishes for about 10 minutes. This helps the butter layers stay firm so the pastries puff properly in the oven.

Bake for 20–25 minutes, until the pastry is puffed and deeply golden. Let the danishes cool on the baking sheet for at least 10 minutes.

For the optional glaze, whisk the confectioners’ sugar with 1–2 tablespoons of milk or lemon juice until smooth. Drizzle it over the cooled danishes before serving.