Step 1: Make the Dough

1. In a large mixing bowl, whisk together the flour, sugar, and salt.
2. Add the cold, cubed butter to the flour mixture. Use a pastry cutter or your fingertips to cut the butter into the flour until the mixture resembles coarse crumbs.
3. Gradually add the ice water, 1 tablespoon at a time, mixing with a fork or your hands until the dough comes together. Be careful not to overwork the dough.
4. Form the dough into a disc, wrap it in plastic wrap, and refrigerate for at least 30 minutes. This step ensures a flaky crust.

Step 2: Prepare the Apple Filling

1. In a medium skillet over medium heat, melt the butter.
2. Add the diced apples, brown sugar, cinnamon, and nutmeg to the skillet. Stir to coat the apples evenly in the sugar and spices.
3. Cook for 5-7 minutes, stirring occasionally, until the apples are tender but not mushy.
4. In a small bowl, mix the cornstarch and water to create a slurry. Add this to the skillet and cook for an additional 1-2 minutes, allowing the mixture to thicken.
5. Remove from heat, stir in the vanilla extract, and let the filling cool completely before assembling the empanadas.

Step 3: Assemble the Empanadas

1. Preheat your oven to 375°F (190°C) and line a baking sheet with parchment paper.
2. Roll out the chilled dough on a floured surface to about ⅛-inch thickness. Use a 4-5 inch round cutter to cut out circles of dough. Re-roll any scraps to make additional rounds.
3. Spoon about 1-2 tablespoons of the cooled apple filling into the center of each dough circle. Drizzle a small amount of caramel sauce over the filling.
4. Fold the dough over the filling to create a half-moon shape. Use a fork to crimp the edges and seal the empanadas.
5. In a small bowl, whisk together the egg and milk to create an egg wash. Brush the tops of the empanadas with the egg wash and sprinkle with granulated sugar.

Step 4: Bake the Empanadas

1. Place the empanadas on the prepared baking sheet, leaving about 1 inch of space between each.
2. Bake for 20-25 minutes, or until the empanadas are golden brown.
3. Remove from the oven and let them cool for 5 minutes before serving.