Preheat oven to 350F. Line a large baking sheet with parchment paper.
In a large mixing bowl, beat butter, brown sugar, and granulated sugar until smooth. Add egg and mix until well combined. Add pumpkin puree and vanilla, mixing until combined. Add pumpkin pie spice, baking powder, baking soda, and salt; mix until everything is combined. Add in flour and combine until flour is well incorporated (do not overmix)
Drop by rounded tablespoonfuls (or use a cookie scoop) onto prepared baking sheets. Bake 10 to 12 minutes, or until cookies are set. Allow cooling completely before frosting.
Using a stand mixer, beat cream cheese and butter at medium-high speed until creamy. Add confectioners’ sugar and vanilla, mixing until soft and creamy. Generously frost the cooled cookies, and if desired, sprinkle the tops of the cookies with a pumpkin pie spice. Store in the refrigerator on plates loosely covered with aluminum foil in the refrigerator. These are best eaten the same day or within 2 days.
To freeze, store uncooked cookie dough balls for up to 3 months.