Preheat the oven to 375°F (190°C). Line a baking sheet with parchment paper.
In a bowl, combine the pumpkin puree, brown sugar, cinnamon, and nutmeg (or pumpkin pie spice). Mix until smooth and well combined.
Roll out the store-bought puff pastry on a lightly floured surface. Cut the sheet into 4 squares (or 6 if you want smaller turnovers).
Place a spoonful of the pumpkin filling in the center of each pastry square. Fold the pastry over to form a triangle and press the edges with a fork to seal.
In a small bowl, whisk together the egg and a tablespoon of water. Brush the egg wash over the tops of the turnovers to give them a golden color when baked.
Place the turnovers on the prepared baking sheet and bake for 20-25 minutes or until golden and puffed.