Preheat the oven to 400F. Line a large rimmed baking sheet with parchment paper or a silicone baking mat. Set aside.
Melt butter in a large skillet over medium heat.
Add apples to skillet and cook for about 2-3 minutes. Apples should be beginning to soften.
Stir in brown sugar and cinnamon.
Cook, stirring regularly, for 2 more minutes.
Stir together cornstarch and 1 tablespoon water.
Add to the skillet, and mix well.
Cook for another minute, or until sauce has thickened.
Remove from heat to cool slightly.
Unfold puff pastry sheets, and press sheets back together to fix any cracks.
Cut each sheet into 4 squares.
Spoon apples onto the center of each squares. Don't spoon the accumulated juices in.
Fold over from corner to corner into a triangle shape.
Crimp edges together to seal.
Place turnovers on a baking sheet, leaving about 1 inch between them.
Sprinkle the tops with the sugar. You can also dust the tops with cinnamon if desired.
Bake for 20-30 minutes or until turnovers are puffed and light golden brown.
Notes
Baking apples like Granny Smith or Jonagold work well in this recipe.
To make a glaze for the turnovers, whisk together 1/2 cup powdered sugar, 1 tablespoon melted butter, and enough milk to make a drizzle-able glaze. Drizzle it over the baked and cooled turnovers.