Combine the pumpkin puree, softened cream cheese, ground cinnamon, grated nutmeg, salt, vanilla extract, egg, and packed brown sugar in a medium bowl. Whisk everything together until the mixture is completely smooth and creamy. Set the filling aside while you work on the dough.

Whisk the all-purpose flour, cane sugar, and salt together in a large bowl. Add the very cold cubed butter and quickly rub it into the flour mixture with your fingertips until it resembles small peas. Sprinkle in 2 tablespoons of ice water and fluff with two forks. Keep adding ice water one tablespoon at a time, fluffing after each addition, until you have used 5 tablespoons total. If the dough still looks dry, add a little more water in 1/2 teaspoon increments until the dough holds together as shaggy crumbs.

Turn the dough out onto a work surface and gently knead it just a few times until it comes together into a cohesive mass. Divide the dough into two equal pieces, flatten each into a disk, wrap tightly in plastic wrap, and refrigerate for at least 1 hour. When ready to roll, remove one disk from the fridge and let it rest for about 10 minutes. On a well-floured surface, roll the dough out to approximately 1/8 inch thick. Using a 4-inch round cutter or glass, cut out 8 rounds and return them to the refrigerator. Repeat with the second dough disk to yield 16 rounds total. Keeping the cut rounds chilled ensures the butter stays cold for a flakier bake.

Place roughly 2 1/2 teaspoons of the pumpkin and cream cheese filling in the center of each dough round. Lightly moisten the edges of the rounds with water, then fold each one over the filling to form a half-moon shape. Press the edges together firmly to seal, then use a fork to crimp all the way around, making sure there are no gaps that could allow filling to escape during baking.

Line a baking sheet with parchment paper and arrange the shaped empanadas on it in a single layer. Transfer the pan to the freezer and chill the empanadas for 15 minutes to help them hold their shape. Meanwhile, preheat your oven to 375°F (190°C).

Remove the empanadas from the freezer. Whisk together 1 egg and 1 tablespoon of water in a small bowl to make the egg wash, then brush it evenly over each empanada. In a separate small bowl, stir together the cane sugar and ground cinnamon, then sprinkle the mixture generously over the tops of all the empanadas.

Bake the empanadas for about 20 minutes, or until they are puffed and deep golden brown. Let them cool for a few minutes before serving. They are wonderful eaten warm, but equally good at room temperature.