Cream the butter and powdered sugar in a large bowl with a hand mixer on medium speed for 2 to 3 minutes until light and fluffy.

Mix in the pumpkin puree and vanilla extract just until combined, keeping mixing short and gentle.

In a separate bowl, whisk together the flour, pumpkin pie spice, and salt. Add the dry mixture to the butter mixture and mix on low speed until a soft dough forms and no dry flour remains.

Shape the dough into a log about 2 inches in diameter, wrap it in plastic wrap, and refrigerate for at least 1 hour.

Preheat the oven to 325°F and line two baking sheets with parchment paper.

Slice the chilled log into rounds about 1/3 inch thick and place them on the baking sheets about 1 inch apart.

Bake for 14-16 minutes until the edges are barely golden. Let the cookies cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.

If using icing, whisk together the powdered sugar, milk, and cinnamon until smooth and drizzly. Spoon it over the cooled cookies or use it to stack 2 to 3 cookies together.