To make the cookies: Preheat oven to 350 F degrees. Cover a large cookie sheet with parchment paper and set aside. In a large bowl, beat butter with an electric mixer on medium to high speed for 1 minute. Add sugar, baking powder, baking soda, salt, cinnamon, and pumpkin pie spice and beat until well combined. Add egg and 1 vanilla and beat well then add pumpkin and beat until combined. Add flour and beat on medium speed until completely incorporated. The dough will be wet.
Drop dough by heaping tablespoon a few inches apart on prepared cookie sheet. Bake for 10 to 12 minutes or until tops are set and no longer look wet. Allow to cool on cookie sheet for a few minutes then transfer to wire rack to cool completely before frosting.
To make the cream cheese frosting: In a large bowl, beat together butter and cream cheese until smooth with no lumps. Sift in the powdered sugar and add vanilla. Beat until smooth and creamy, about 2 minutes. Once cookies are cool, frost with cream cheese frosting. Serve immediately or keep in an air tight container in the refrigerator for up to one week.