Preheat oven to 350°F and line baking sheets with parchment.
In a stand mixer, combine sugars and oil until smooth.
Add pumpkin purée, eggs, and vanilla; mix to combine. Gradually mix in flour, salt, baking soda, baking powder, and pumpkin pie spice until just combined.
Scoop batter onto baking sheets and bake for 10–12 minutes. Let cookies cool completely.
For the frosting, beat softened butter, then mix in cream cheese until smooth.
Add powdered sugar and vanilla; beat until fluffy.
Pipe frosting onto half the cookies, then top with a second cookie to create sandwiches.
Store in the fridge for up to 3 days.