Preheat the oven to 400˚F and line 2 sheet trays with parchment paper. Set aside.
In a bowl, combine the pumpkin puree, egg, brown sugar, vanilla extract, flour, cinnamon, ginger, cloves, nutmeg, and salt. Mix with a whisk until everything is incorporated.
Unfold the puff pastry and roll a rolling pin over it to flatten the seams. Cut the puff pastry into 9 squares.
Take one square place it in front of you so that it looks like a diamond. Fold and align the bottom corner up to the top corner. You should now have a triangle. Use a paring knife to cut a slit 1/4 inch away from the border on the right and left sides. Make sure that the left and right slits do not meet at the top, leaving a space of the pastry between them. Unfold the pastry so that it looks like a diamond again. Fold the outer edge that is now cut over to the left side so that the corner meets the inner left corner. Then fold the outer edge on the left side so that the corner meets the right corner. Please use the photos in the blog post as a reference for clarification.
Place the folded pastries onto a sheet tray, lined with parchment paper, spacing them 2 inches apart. If you notice your pastry has gotten soft and sticky, place them into the freezer for about 5-8 minutes. Spoon about 1 1/2 tbsp of the pumpkin mixture into the center of the pastries.
In a small bowl whisk together the 1/2 egg and 1 tbsp of water to form an egg wash. Use a pastry brush to brush the egg wash onto the borders of the puff pastry. Sprinkle the borders with the demerara sugar.
Bake the pastries for 12-15 minutes. The pastries should have puffed and turned golden brown. Transfer them from the tray onto a wire rack to cool.