Place the flour, sugar, and salt in a large mixing bowl and whisk until evenly combined.
Add the chilled, cubed butter and gently toss until the pieces are coated with the flour mixture.
Using a pastry cutter or your fingertips, work the butter into the dry ingredients until the mixture develops a coarse, crumb-like texture.
Add the ice-cold water gradually, approximately 1 tablespoon at a time, while lightly stirring with a fork. Continue only until the flour is moistened and the dough begins to hold together.
Gather the dough into a ball, cover it securely with plastic wrap or parchment paper, and refrigerate for no less than 30 minutes, or until thoroughly chilled and firm. Keeping the dough cold is important for producing a flaky pastry.
Add the sugar, cinnamon, salt, cornstarch, and water to a small saucepan and stir to combine.
Set the pan over medium heat and bring the mixture to a boil, stirring continuously. Cook until the liquid becomes thick and glossy.
Remove the saucepan from the heat. Fold in the finely diced apples along with the lemon juice, when using.
Set the filling aside and allow it to cool completely before proceeding with assembly.
Line a baking sheet with parchment paper.
Lightly flour a work surface and roll out half of the refrigerated pastry dough to approximately 1/8-inch thickness.
Using a sharp knife, pizza cutter, or cookie cutter, divide the pastry into circles, squares, or rectangles as desired.
Spoon approximately 1 to 2 tablespoons of the cooled apple mixture into the center of each piece of pastry.
Fold the dough over the filling to form individual turnovers. Press the edges together, then crimp them securely with the tines of a fork.
Arrange the completed turnovers on the parchment-lined baking sheet.
Place the baking sheet in the freezer for 20 minutes or refrigerate it for 30 minutes.
This additional chilling period firms the pastry and helps the turnovers retain their shape during baking.
Preheat the oven to 400°F (200°C).
Whisk the egg and milk together in a small bowl until evenly blended.
Remove the chilled turnovers from the refrigerator or freezer and lightly coat their tops with the egg wash.
With a sharp knife, make several small slits in the top of each turnover so steam can escape while they bake.
Sprinkle the surfaces generously with the remaining sugar to create a crisp, sparkling finish.
Transfer the turnovers to the preheated 400°F (200°C) oven and bake for 10 minutes.
Reduce the oven temperature to 375°F (190°C) and continue baking for another 18 to 20 minutes, or until the pastry is crisp and evenly golden brown.
Because baking time may vary depending on the thickness of the rolled pastry, monitor the turnovers closely during the final few minutes.
Remove the baked apple turnovers from the oven and allow them to cool slightly before serving.
Serve warm or at room temperature.