Preheat your oven to 375°F (190°C). Lightly butter a 9×13-inch baking dish or spray with nonstick spray.

In a medium saucepan, combine 1 cup granulated sugar, 1 cup water, 1/2 cup unsalted butter, 1/2 teaspoon ground cinnamon, 1/4 teaspoon salt, and 1 teaspoon vanilla extract. Set over medium heat and cook, stirring occasionally, until the butter melts and the mixture just begins to simmer. Do not fully boil. Once smooth and hot, remove from heat and set aside; it will thicken as it bakes.

Peel the 4 apples, then cut each one in half from top to bottom and remove the core. You should have 8 apple halves. Pat them dry with paper towels so the sugar and spices cling well.

In a small bowl, stir together 4 tablespoons light brown sugar, 2 teaspoons ground cinnamon, 1/4 teaspoon ground nutmeg (if using), a pinch of salt, and 1 teaspoon vanilla extract. The mixture will be slightly pasty.

Place the apple halves cut side up on a cutting board. Spoon the cinnamon-sugar mixture into the core cavities, dividing evenly among the 8 halves. Top each half with a piece of the cold butter (8 pieces total).

Unroll each pie crust onto a lightly floured surface. Cut each crust into 4 equal wedges (like a pizza) to make 8 pieces of dough. If the dough is too soft or sticky, chill it in the refrigerator for 5–10 minutes before using.

Place one apple half, cut side down, near the wide end of a dough wedge. Bring the sides of the dough up and around the apple, pressing and pinching to seal at the top. Repeat with the remaining apple halves and dough wedges.

Place each wrapped dumpling, seam side up, in the prepared baking dish, leaving a little space between each one to allow for puffing.

Stir the warm cinnamon syrup and carefully pour it around the dumplings in the pan, avoiding pouring directly over the tops so the egg wash will stick later.

Brush the tops of the dumplings with the beaten egg and water mixture. In a small bowl, combine 2 tablespoons sugar and 1/2 teaspoon cinnamon, then sprinkle generously over the tops of the dumplings.

Bake at 375°F (190°C) for 35–45 minutes, until the pastry is deep golden brown, the syrup is bubbling around the dumplings, and a knife slides easily into the apples. If the tops brown too quickly, tent loosely with foil for the last 10 minutes.

Let the dumplings rest for 10–15 minutes so the sauce can thicken slightly. Serve warm in shallow bowls, spooning some syrup over each dumpling. Top with vanilla ice cream or whipped cream, if desired.