Preheat your oven to 375°F and lightly grease a 9x13-inch baking dish with butter or cooking spray.
Make the pastry dough by combining the flour and salt in a large bowl.
Cut in the cold cubed butter with a pastry cutter or two forks until the mixture resembles coarse crumbs with pea-sized butter pieces.
Gradually add the ice water to the flour mixture, stirring with a fork until the dough just comes together.
Gently knead the dough into a smooth ball on a floured surface before wrapping in plastic and refrigerating for 30 minutes.
Roll the chilled dough on a floured surface into a large rectangle about 1/8-inch thick, then cut into six equal squares large enough to wrap around each apple completely.
Place one apple in the center of each dough square, fill the core cavity with a mixture of 2 tablespoons granulated sugar, cinnamon, nutmeg, and 2 teaspoons softened butter per apple, then bring the dough corners up over the apple, pinching the seams together to seal completely.
Arrange the wrapped apples in the prepared baking dish.
Brush the pastry with beaten egg.
Pour a syrup made by boiling 1 cup water with remaining granulated sugar and brown sugar over the dumplings.
Bake for 45 minutes until golden brown and apples are tender when pierced with a knife.