In a large bowl or stand mixer, cream together the softened vegan butter, pumpkin puree, brown sugar, cane sugar, and vanilla extract until light and fluffy.
In a separate bowl, sift together the flour, baking powder, pumpkin pie spice, cornstarch, and salt. Whisk to combine.
Gradually add the dry ingredients to the wet ingredients and mix on low until a soft dough forms. Do not overmix.
Cover the bowl and refrigerate for at least 30 minutes (or overnight). Do not skip this step.
Preheat oven to 350°F (177°C). Line a baking sheet with parchment paper.
For Halloween shapes: Place half the chilled dough between two sheets of parchment paper and roll to 1/4-inch thick. Cut out shapes with cookie cutters. Transfer to the baking sheet 2 inches apart. If dough is too soft, freeze for 10 minutes before transferring. Bake 6–8 minutes.
For round cookies: Scoop 2-inch balls of dough onto the prepared pan, spacing 3 inches apart. Bake 10–12 minutes.
Remove from oven. Cookies will look slightly underdone - that's correct. They firm up as they cool.
While warm, sprinkle cinnamon sugar over all the cookies.
Allow to cool completely on the pan before removing or storing.