Place the pumpkin puree on a paper towel and let all the moisture drain. Repeat with dry paper towels until all the moisture is gone.
In a large mixing bowl, beat the softened butter, and brown sugar until light and fluffy, about 3 to 4 minutes.
Beat in the pumpkin puree, egg, and vanilla extract until smooth.
In a separate bowl, whisk together the flour, cornstarch, baking soda, salt, and pumpkin pie spice.
Gradually mix the dry ingredients into the wet ingredients just until a soft dough forms.
Chill the Cookie Dough: Divide the dough in half and shape each half into a disk. Wrap tightly and refrigerate for at least 45 minutes.
Preheat the oven to 350°F.
Line baking sheets with parchment paper.
Lightly flour your work surface and roll the dough to about 1/4-inch thickness.
Cut the dough with a pumpkin-shaped cookie cutter and place the cookies about 2 inches apart on the prepared baking sheets.
For the sharpest edges, refrigerate the cut cookies on the baking sheets for 10 to 15 minutes before baking.
Bake for 10 to 12 minutes, or until the cookies are set and the bottoms are just beginning to turn lightly golden.
Cool on the baking sheets for 5 minutes before transferring to a wire rack to cool completely.