Add the flour, powdered sugar, pumpkin pie spice, and salt to the bowl of a food processor. Mix on high until well combined.
Cut the cold butter into tablespoon-sized pieces.
Add 5-6 pieces of butter to the dry ingredients and turn the processor on high until mixed in. Repeat this process until all of the butter has been added.
Add the pumpkin puree into the bowl of the food processor and blend until the dough starts to form.*
Roll the dough into a flat disc, cover in plastic wrap, and refrigerate for at least an hour.
Preheat the oven to 325° and line baking sheets with parchment paper or silicone mats.
Roll the dough out to ¼" thickness and cut out circles.
Add cookies to lined baking sheet. If stamping them, do so at this point.
Bake for 14-16 minutes until the bottom edges are slightly golden brown and the top no longer looks shiny.
Remove from the oven and allow the cookies to set on the tray for at least 10. Once set and mostly cooled, move to a wire cooling rack to completely cool.