Preheat oven to 375°F (190°C). Lightly grease a pie dish.
In a mixing bowl, toss apples with brown sugar, cinnamon, nutmeg, lemon juice, cornstarch, and vanilla extract. Let sit for 10 minutes to release juices.
Roll out one puff pastry sheet and fit it into the bottom of the pie dish, trimming any excess.
Pour the apple mixture into the pastry-lined pie dish, spreading evenly.
Place the second sheet of puff pastry over the apples. Seal and crimp the edges, then cut small vents on top to allow steam to escape.
Brush the top crust with the beaten egg wash for a golden finish.
Bake for 40–50 minutes, until the crust is golden brown and the filling is bubbling.
Let cool for at least 20 minutes before slicing. Serve warm with vanilla ice cream or caramel sauce.
Notes
For added crunch, sprinkle turbinado sugar on the crust before baking. Try mixing apple varieties for a more complex flavor.