In a medium-sized mixing bowl, combine pumpkin puree, brown sugar, granulated sugar, ground cinnamon, ground nutmeg, and vanilla extract. Stir until smooth and well-blended. Chill for 15-30 minutes.

Preheat the oven to 400°F (200°C). Roll out the thawed puff pastry sheets on a floured surface to approximately 10x12 inches each. Slice each sheet into 6 rectangles of about 4x5 inches.

Place about a tablespoon of the chilled pumpkin filling in the center of each pastry rectangle. Fold over to cover the filling, sealing the edges. Twist the ends to form spirals.

Line a baking sheet with parchment paper. Arrange the twisted pastries, brush the tops with egg wash, and bake for 15-20 minutes or until golden.

Remove from the oven and cool on a wire rack for about 5 minutes before serving.