For the crust
1 box (2 sheets) puff pastry, thawed
1 egg, lightly beaten
For the filling
2 baking apples, peeled and cored (I've used both Honeycrisp and Gala)
3 tablespoons unsalted butter, divided
1/2 cup packed dark brown sugar
1 tablespoon all-purpose flour
2 teaspoons freshly squeezed lemon juice
1/2 teaspoon ground cinnamon
Pinch of kosher salt
For the icing
4 ounces cream cheese
1 cup powdered sugar
1 teaspoon pure vanilla extract
For the filling, grate the apples on the large holes of a box grater. In a heavy saucepan, melt 2 tablespoons of butter over medium heat. Add the grated apples and toss until coated. Add the brown sugar, flour, lemon juice, cinnamon, and salt. Continue cooking, stirring frequently, until the mixture is thick and syrupy, about 10 minutes. Remove from heat and stir in remaining tablespoon of (chilled) butter. Cool the filling to room temperature.
Preheat oven to 375°
Unfold one sheet of puff pastry onto a well-floured work surface, continuing to flour the work surface as necessary throughout the rolling/trimming process. Roll out the pastry to form a very large, thin rectangle. Trim the edges of the dough to make a 10 x 14 inch rectangle. Using a sharp knife or pizza cutter, cut out 8 (3-1/2 x 5 inch) rectangles.
Transfer 4 of the rectangles to a parchment-lined baking sheet. Spoon a scant 2 tablespoons of the apple filling into the center of these. Spread the filling evenly across the pastry, leaving a 1/2-inch space clean around the perimeter. Brush the other 4 rectangles with the egg wash. Place an egg washed rectangle directly over a filled rectangle (egg wash touching the filling). Using your fingertips, firmly seal the edges on all sides (dip your fingers in a little flour if the dough is sticky). Continue with the rest of the rectangles. Prick each strudel all over with a fork, including the sealed edges. Repeat this step with the second sheet of puff pastry.
Bake the strudels until puffed and golden brown, about 35 minutes. Cool for 15 minutes on a wire rack before eating. Frost and serve warm.
For the icing, beat the cream cheese, powdered sugar, and vanilla extract together. Spoon the glaze into a small Ziploc bag and seal. Snip a small hole in one corner of the bag and pipe desired amount onto each pastry.