Prepare the Filling: Combine pumpkin puree, brown sugar, granulated sugar, ground cinnamon, ground nutmeg, and vanilla extract in a mixing bowl. Stir until smooth, cover, and chill for 15-30 minutes.

Roll Out the Pastry: Dust a clean countertop with flour and roll each sheet of puff pastry to 10x12 inches and about an eighth of an inch thick.

Cut the Pastry: Slice each sheet into six equal rectangles, about 4x5 inches, ensuring clean edges.

Fill the Pastry: Place a tablespoon of chilled filling in the center of each rectangle. Fold over the top, seal edges, and twist the ends.

Preheat the Oven: Set oven to 400°F (200°C) and line a baking sheet with parchment paper.

Brush with Egg Wash: Whisk egg and water together for egg wash, then lightly coat tops of twists.

Bake the Twists: Place twists on the baking sheet and bake for 15-20 minutes until golden brown.

Cool Before Serving: Let the twists cool on a wire rack for a few minutes before serving.