Place the chopped apples and 3 tablespoons butter in a medium saucepan. Add the brown sugar, cinnamon and salt to the saucepan. Bring the mixture to a boil over medium-high heat and then reduce the heat to a simmer. Cook until the apples are tender, about 5 minutes. Remove from heat and allow to cool.

In a mixing bowl, combine the softened cream cheese, 1/3 cup white sugar, cardamom, and vanilla.  Set aside and prepare the puff pastry.

Preheat oven to 400 degrees.

Carefully unfold the sheet of puff pastry and place on top of a sheet of parchment paper. Lightly roll with a floured rolling pin until it is ¼ inch thick.  Transfer the puff pastry (including the parchment paper) to a baking tray.

Down the center of the puff pastry, add the cream cheese mixture.

Arrange the cooled apple filling on top of the cream cheese mixture.

Using a sharp knife or a pizza cutter, begin slicing diagonal strips down both sides of the puff pastry, being careful to leave a ½ inch border around the cream cheese and apple slices.

There should be around 8 to 10 strips depending on the length of the pastry braid.

Starting at one end, begin folding the strips up and over the apples, alternating sides to form a braid with the dough.

Continue to fold the dough strips until the apples are covered.  Tuck the end strips under.

In a small bowl, beat with a fork one egg white.

Brush the pastry braid with the egg white and sprinkle the top with 1 to 2 tablespoons white sugar.

Place in the oven and bake for 20 to 25 minutes or until golden brown.Allow to cool slightly before drizzling with glaze.

To make the glaze, heat the 2 tablespoons of butter in a saucepan over medium heat. Heat until it gives off a nutty fragrance, about 5 minutes, to make a “brown butter.” Remove from heat and transfer to a bowl to cool slightly. Stir in the powdered sugar and vanilla. Mix in the milk a teaspoon at a time until the glaze is thin enough to drizzle. Drizzle the apple braid with the glaze; slice and serve!

Meanwhile, make the filling. Place the chopped apples and 3 tablespoons butter in a medium saucepan. Combine 1/2 cup brown sugar, 1 1/2 tablespoons flour, 1/4 teaspoon salt, cinnamon, and nutmeg and mix well; add the mixture to the saucepan. Bring the mixture to a boil over medium-high heat and then reduce the heat to a simmer. Cook until the apples are tender, about 5 minutes. Remove from heat and allow the filling to cool.

To make the glaze, heat the 2 tablespoons of butter in a saucepan over medium heat. Swirl the pan occasionally and cook until the milk solids turn brown and give off a nutty fragrance, about 5 minutes. Remove from heat and transfer to a bowl to cool slightly. Stir in the confectioners' sugar and vanilla. Mix in the milk a teaspoon at a time until the glaze is thin enough to drizzle. Decorate the Danish braid with the glaze.