1. In a stand mixer, cream together vegan butter, pumpkin puree, brown sugar, granulated sugar, and vanilla until light and fluffy.
2. In a separate bowl, whisk together flour, baking powder, pumpkin pie spice, cornstarch, and salt.
3. Gradually add dry ingredients to wet, mixing on low speed until just combined.
4. Cover and refrigerate dough for 30 minutes.
5. Preheat oven to 350°F (177°C).
6. Scoop 2-inch balls of dough, flatten into disks, and place on a parchment-lined baking sheet.
7. Bake for 10-12 minutes. Let cool completely on the baking sheet.
8. For frosting, whip all frosting ingredients until combined. Pipe onto cooled cookies.