To a medium saucepan, add the apples, butter, brown sugar, cinnamon, cornstarch, stir to combine, and heat over medium-high heat until the mixture reaches a boil. It will seem very dry and crumbly until the apples soften and release their juices, but just keep stirring and tossing.
After mixture reaches a boil, turn the heat to low and allow mixture to simmer for about 5 minutes. Set aside to cool momentarily while you prepare the puff pastry.
Preheat oven to 400F.
Cut each large sheet of puff pastry into 4 squares so that you have a total of 8 small squares. If necessary, add flour to your work surface so puff pastry doesn’t stick.
Add about 1/4 cup filling to the center of each square, fold on the diagonal, crimp edges very well with a fork to prevent leakage, and transfer to a half-sheet pan lined with a Silpat, or sprayed well with cooking spray.*
Arrange the 8 turnovers on pan so edges aren’t touching. It’s okay if they don’t have much space in between them because they rise rather than spreading much; set aside.
To a small bowl, add the egg, water, beat with a fork, and lightly brush each turnover with eggwash; discard remainder.
Generously sprinkle each turnover with granulated sugar. I use about 1 teaspoon per turnover.
Bake for about 30 to 34 minutes, or until turnovers are lightly golden brown on top. Watch the undersides because they will brown much faster than the top and be more prone to burning and any filling that leaks will also be very prone to burning.
Serve immediately with any reserved filling mixture if desired. Turnovers are best warm and fresh but will keep airtight for up to 4 days.