In a medium bowl, combine pumpkin puree, brown sugar, granulated sugar, ground cinnamon, ground nutmeg, and vanilla extract. Mix until smooth and thoroughly blended, then cover and chill in the refrigerator for 15 to 30 minutes.

Preheat your oven to 400°F (200°C) and line a baking sheet with parchment paper. On a lightly floured surface, roll out each sheet of thawed puff pastry to approximately 10x12 inches. Cut each sheet into six equal rectangles.

Take one rectangle of puff pastry and place about 1 tablespoon of the chilled pumpkin filling in the center. Fold the pastry over to create a pocket, sealing the edges by pressing firmly.

Arrange the assembled twists on the prepared baking sheet, leaving space between each one. Brush the tops of the twists generously with the egg wash. Bake in the preheated oven for 15 to 20 minutes.

Once baked, remove from the oven and let cool on a wire rack for about 5 minutes. Serve warm.