Preheat Oven: 400°F (200°C). Thaw puff pastry until pliable.
Make the Apple Filling: In a large bowl, combine sliced apples, brown sugar, granulated sugar, cinnamon, nutmeg, vanilla, and lemon juice. Stir to coat.
Add Flour: Stir in flour to evenly distribute (helps thicken the filling).
Prepare Bottom Crust: Lay one puff pastry sheet in a 9-inch pie pan, press gently to fit.
Add Filling: Pour apple mixture over the bottom crust.
Top Crust: Place the second puff pastry sheet on top; press edges to seal.
Vent and Egg Wash: Cut small slits for steam, brush with egg wash, sprinkle with sugar.
Bake: 1 hour to 1 hour 15 minutes, until crust is golden brown and filling juices bubble.
Cool and Serve: Let pie cool for about 30 minutes before slicing.