Preheat the oven to 400° F.
Cut the puff pastry sheet into thirds
Line a baking sheet with parchment paper and place the3 puff pastry rectangles on the baking sheet. Place in the refrigerator while preparing the apples.
Halve the apples and remove the center. Make sure to slice very thinly, either by hand or with a mandoline. I do not peel the apples, but if you prefer, you can peel them before slicing.
Place the sliced apples in a large bowl. Add the brown sugar, vanilla extract, cinnamon, allspice, melted butter, and salt. Mix with a spoon to thoroughly coat the apples.
Remove the baking sheet with the puff pastry rectangles from the fridge.
Arrange apple slices on the pastry sheets, overlapping as you go.
Sprinkle any additional sugar mixture evenly over the apples.
Beat the egg with a tablespoon of water.
Coat the outer edges of the puff pastry with the beaten egg using a pastry brush.
Sprinkle the turbinado sugar on the edges of the puff pastry.
Bake for 20 minutes or until the apples are tender and the pastry is golden brown and puffy.
Remove from the baking sheet, place on a serving tray, and slice each rectangle tart into three portions.
Serve immediately or at room temperature with your favorite toppings.