Preheat oven to 400°F. Line a large baking sheet with parchment paper.

In a medium saucepan, combine diced apples, granulated sugar, brown sugar, cinnamon, and lemon juice. Cook over medium heat for 3-4 minutes until apples begin to soften.

In a small bowl, whisk together cornstarch and water to make a slurry. Stir the slurry into the apple mixture and continue cooking for 2-3 minutes, stirring frequently, until thickened. Stir in butter and remove from heat. Let cool completely.

Roll out each puff pastry sheet on a lightly floured surface. Cut each sheet into 4 squares (8 total).

Place 2 tablespoons of cooled apple filling in the center of each square. Brush edges with beaten egg. Fold diagonally to form a triangle and press edges with a fork to seal.

Place turnovers on prepared baking sheet. Brush tops with beaten egg and cut 2-3 small slits in the top of each for steam vents.

Bake for 20-25 minutes or until golden brown and puffed.

If making glaze: Whisk together powdered sugar, milk, and vanilla until smooth. Drizzle over turnovers after cooling 5 minutes.

Serve warm.