In a large bowl, combine the flour, salt, and granulated sugar. Mix well.

Add the chilled, cubed butter to the flour mixture. Use a pastry cutter or your fingers to blend until the mixture resembles coarse crumbs.

Gradually add the ice water, one tablespoon at a time, mixing until the dough comes together. Do not overmix.

Shape the dough into a ball, wrap it in plastic wrap, and refrigerate for at least 1 hour.

Preheat your oven to 400°F (200°C) and line a baking sheet with parchment paper.

In a medium bowl, combine the diced apples, cinnamon, nutmeg, lemon juice, and brown sugar. Mix until the apples are well coated.

On a floured surface, roll out the chilled dough to a thickness of about 1/8 inch. Cut the dough into squares (approximately 4-5 inches).

Place a spoonful of the apple mixture in the center of each square.

Fold the dough over to form a triangle and press the edges to seal. Use a fork to crimp the edges for a decorative touch.

Brush the tops of the turnovers with the beaten egg.

Bake in the preheated oven for 20-25 minutes, or until golden brown.

Remove from the oven and let cool slightly. Dust with powdered sugar, if desired, before serving.