Mix the flour and a pinch of salt in a large bowl. Add the cold butter cubes and use a pastry cutter or your fingers to blend until the mixture resembles coarse crumbs.
Gradually pour in the ice water, a tablespoon at a time, mixing gently until the dough starts to come together. Stop adding water once the dough holds when pressed lightly.
Shape the dough into a disk, wrap it in plastic wrap, and refrigerate for about 30 minutes to let the gluten relax.
While the dough chills, prepare the filling: toss the diced apples with brown sugar and cinnamon until evenly coated.
Preheat your oven to 400°F (200°C). Remove the dough from the fridge and roll it out on a lightly floured surface to about 1/8 inch thickness.
Use a round cutter or a glass to cut out circles approximately 4 inches in diameter each.
Spoon a tablespoon of apple filling onto one side of each dough circle, leaving a small border around the edge.
Fold each circle over to enclose the filling, then crimp the edges with a fork to seal tightly.
Place the assembled pies on a baking sheet lined with parchment paper. Brush the tops with beaten egg for a shiny, golden crust.
Bake for about 20-25 minutes, or until the pies are golden brown and bubbly. Remove from the oven and let cool slightly.
Drizzle caramel sauce over the warm pies before serving for an extra touch of sweetness and richness.
