Preheat oven to 400°F. Thaw frozen puff pastry per package instructions, or keep refrigerated as directed.
In a sauté pan over medium heat, melt the butter. Once melted, add the apples, brown sugar, and cinnamon. Stir to coat apples. Cook for 5-10 minutes, until the apples are softened, but not mushy. Set aside.
In a mixing bowl, beat together softened cream cheese and granulated sugar until smooth. Beat in egg yolk, vanilla, and salt.
Unroll puff pastry onto a sheet of parchment paper. Place another sheet of parchment on top and gently roll out the pastry to about 10x16 inches. Remove the top sheet of parchment and transfer the pastry to a baking sheet.
Spread cream cheese mixture lengthwise down the middle of the pastry, leaving about 2 inches on each side and 1 inch on the bottom. Spoon apples on top.
Use a paring knife, kitchen shears, or pizza cutter to cut diagonal strips on both sides of the pastry.
Starting at the top, fold the strips over toward the middle, stretching gently to overlap pieces. Work your way down to the bottom, then tuck in bottom and top of pastry.
Brush egg white over the pastry. Sprinkle on coarse sugar if using.
Bake for 25-30 minutes, until golden brown. Let cool before serving.
Optional topping
Once pastry is cooled to room temperature, whisk together powdered sugar and milk in a bowl until you have a smooth, pourable glaze.
Drizzle over the top of the apple danish. If using almonds, sprinkle on top.