Keep the dough refrigerated until you are ready to roll and assemble the Danish. The dough should remain cold throughout shaping so the butter layers stay intact.

Beat the cream cheese and sugar together just until smooth.

Add the flour and mix until incorporated.

Add the egg yolk, vanilla extract, and salt and mix just until combined.

Refrigerate the filling while you roll and shape the dough.

Peel the apples if desired, then dice them into small pieces, about ¼ inch. Keeping the pieces small makes the Danish easier to assemble and eat.

Melt the butter in a medium saucepan over medium heat.

Add the apples, brown sugar, cinnamon, nutmeg, lemon juice, and salt. Stir to coat the apples evenly.

Cook for about 7 minutes, stirring occasionally, until the apples are just tender but still hold their shape.

In a small bowl, stir together the cornstarch and water until completely smooth.

Pour the cornstarch mixture into the apples and stir well. Continue cooking for 1 to 2 minutes, stirring constantly, until the juices become thick and glossy.

Transfer the apple filling to a bowl, press plastic wrap directly onto the surface, and allow it to cool completely before assembling the Danish. The filling should be quite thick once cooled rather than saucy. Excess liquid can make the center of the pastries soggy.

Preheat the oven to 350ºF (175ºC). Line a small baking sheet with parchment paper.

In a medium bowl, combine the flour, granulated sugar, brown sugar, cinnamon, and nutmeg.

Add the softened butter and mix with a fork until moist crumbs form. Some larger pieces are fine and will give the streusel more texture.

Spread the streusel in an even layer over the prepared baking sheet.

Bake until lightly golden and crisp, about 10 to 12 minutes, stirring or breaking up the crumbs once during baking if necessary.

Allow the streusel to cool completely. Once cool, break any very large pieces into smaller crumbs. Set aside for topping the baked Danish.

Line two baking sheets with parchment paper.

Place the chilled Danish dough on a lightly floured surface. Roll it to approximately ⅓ inch thick, or slightly thinner.

As you roll, periodically lift the dough from the work surface to make sure it isn't sticking. Add only as much flour as necessary.

If the dough begins shrinking back significantly while you are rolling it, cover it and allow it to rest for 5 to 10 minutes before continuing. This allows the gluten to relax and makes shaping easier.

Using a 3¼-inch round cutter, cut rounds from the dough.

Gather the scraps gently without kneading them together aggressively. Allow them to rest for several minutes before rerolling and cutting additional rounds.

If the rerolled dough continues springing back, give it additional short resting periods between rolling. This helps prevent the rounds from shrinking after they are cut.

Arrange the rounds on the prepared baking sheets, leaving room between each one.

Using a 1½-inch round cutter, gently press an indentation into the center of each round. Do not cut completely through the dough. The indentation creates a space for the fillings while leaving a raised border around the pastry.

Place about 1 tablespoon of cream cheese filling inside the indentation of each Danish.

Spoon about 1 tablespoon of cooled apple filling over the cream cheese.

Keep the filling concentrated toward the center so the outer edge of the pastry remains exposed. I also like to leave a little of the cream cheese visible around the apples so you can see the separate cream cheese and apple layers after baking.

Cover the shaped pastries loosely and let them rise at room temperature for 60 to 90 minutes, or until visibly puffy.

The Danish will not necessarily double in size, but the dough should look noticeably swollen and the laminated layers should become more apparent. The surface of the Danish should feel fluffy and spring back when touched.

Avoid proofing the pastries in a very warm environment. A room temperature of about 72 to 75°F is ideal.

Toward the end of the proofing time, preheat the oven to 400°F (205°C) for at least 30 minutes.

Whisk the egg and milk together to make the egg wash.

Brush the exposed dough lightly with the egg wash, avoiding the cut edges as much as possible.

Bake the Danish at 400°F (205°C) for 8 minutes.

Without opening the oven unnecessarily, reduce the oven temperature to 375°F (190°C) and continue baking for another 10 to 14 minutes, or until the pastries are deeply golden brown.

The bottoms should also be golden and crisp.

Transfer the pastries to a cooling rack and let cool for at least 15 minutes before serving.

Sprinkle the baked streusel generously over the apple filling.

Serve slightly warm or at room temperature.