Preheat oven to 400F (205C) and line two baking sheets with parchment paper. Set aside.
Begin preparing apple filling by combining chopped apples, brown sugar, butter, cornstarch, cinnamon, and salt in a medium-sized saucepan over medium-low heat.
Cook, stirring frequently, until butter is melted. Increase heat to medium and continue to cook, bringing mixture to a simmer. Simmer for 5 minutes, stirring occasionally. Mixture should thicken slightly.
Remove from heat and stir in vanilla extract. Allow mixture to cool before filling pie dough.
Prepare turnover dough: Divide pie crust in half and roll out one half onto a clean, lightly floured surface. Roll into a rectangle between ⅛”-¼” thick.
Using a pizza cutter or sharp knife, neatly cut dough into 4 ½” squares and transfer to a parchment paper lined baking sheet. Repeat with remaining half of dough (note: you will need to gather the scraps and re-roll them out to make all 10 turnovers).
Prepare egg wash by whisking together egg and water. Use a pastry brush to lightly brush the surface of each pastry square with egg wash.
Portion about 1½ Tablespoons of filling just off center of each square. Gently fold over one corner to touch the other to envelope the apple filling. Use the tines of a fork to press and seal the turnovers. Repeat with remaining turnovers.
Brush the outside of each turnover lightly with egg wash and sprinkle with coarse sugar (or regular granulated sugar), if desired. Use a sharp knife to cut 2-3 small slits to vent.
Transfer to center rack of 400F (205C) oven and bake for 25 minutes, until pastry is beginning to turn golden brown.
Allow to cool before topping with icing (if desired).
Whisk together powdered sugar, 1 Tablespoon of milk, and vanilla extract. If mixture is too dry, add additional milk, a splash at a time, until mixture ribbons off the spoon.
Drizzle glaze over cooled turnovers, and enjoy.