Prepare the Apples: Peel, core, and dice the apples into small, evenly sized pieces. Place them in a large mixing bowl and toss with the brown sugar, cinnamon, lemon juice, cornstarch, and vanilla extract until the apples are evenly coated.
Cook the Apple Filling: Melt the butter in a large skillet over medium heat. Add the seasoned apples and cook, stirring occasionally, until they begin to soften and the juices thicken into a lightly syrupy coating. Remove from the heat and let the filling cool completely before assembling the strudel.
Prepare the Puff Pastry: Place the thawed puff pastry on a lightly floured work surface. Gently roll it into a large rectangle, taking care not to overwork the pastry. If it becomes too soft, place it in the refrigerator for a few minutes before continuing.
Add the Apple Filling: Place the cooled apple filling in the center of the pastry, leaving enough room around the edges for folding and sealing. Spread the apples into an even layer without piling the filling too high.
Fold and Seal: Carefully fold the pastry over the apple filling and press the edges firmly together to seal. Transfer the strudel to a parchment-lined baking sheet with the seam facing down.
Add the Egg Wash: Whisk the egg and water together in a small bowl. Brush the egg wash evenly over the entire surface of the strudel. Use a sharp knife to make several small vents across the top to allow steam to escape while baking.
Bake Until Golden: Preheat the oven to 400°F. Bake the strudel for about 30 minutes, or until the pastry is puffed, crisp, and deeply golden brown. Keep an eye on it during the final few minutes to prevent excessive browning.
Cool and Serve: Remove the strudel from the oven and let it rest for about 10 minutes before slicing. Dust with powdered sugar if desired and serve warm with vanilla ice cream, whipped cream, or your favorite topping.