Position the oven racks 3 slots apart, and preheat the oven to 400°F. Line 2 sheet pans with parchment paper.
Place apples, sugar, apple pie spice, lemon juice, and salt in a sauce pot over medium-high heat.
Whisk the cornstarch into the water, and stir into the pot. Bring up to a simmer, and reduce heat to medium-low. Cover, and let cook for 5 minutes.
Remove cover, and simmer for an additional 4 minutes to thicken the apple mixture further. Let cool.
Unroll 1 sheet of puff pastry, and use a pastry cutter to cut 3 lines vertically by 4 lines horizontally to create 12 rectangles. Use a floured rolling pin to roll these rectangles about ⅓ larger, pushing the rolling pin away from you to roll the rectangles into more of a square shape. Repeat with the second sheet of puff pastry.
Place 1 heaping tablespoon of the apple filling in the center of each pastry square. Brush the edges of the pastry with water, fold over, and press hard to fully seal. Use a pastry cutter to trim any uneven excess to make perfectly folded triangles.
Transfer the filled turnovers to the prepared baking sheets, and brush the tops with the beaten egg. Make a ¼-inch-thick cut in the center of each to vent. Sprinkle coarse sugar over each, if desired.
Bake for 12–14 minutes, or until golden brown. Let cool for at least 5 minutes before serving.