Preheat oven to 350 F degrees. Line a large cookie sheet with parchment paper.
In a large bowl with an electric mixer, or bowl of a stand mixer with the paddle attachment, beat butter on medium-high speed for 1 minute. Add sugar, baking powder, baking soda, salt, cinnamon, and pumpkin pie spice and beat until well combined, scraping down the sides of the bowl as needed.
1 cup unsalted butter, 1 cup granulated sugar, 1 teaspoon baking powder, 1 teaspoon baking soda, 1/2 teaspoon salt, 1 teaspoon ground cinnamon, 1 teaspoon pumpkin pie spice
Add the egg and vanilla then beat well, until no yellow streaks remain. Add the pumpkin pure and beat again until incorporated. With the mixer running on low, add the flour turn up to medium speed until completely incorporated. The dough will be slightly wet, this is what you want!
1 large egg, 1 teaspoon vanilla extract, 1 cup canned pumpkin puree, 2 cups all-purpose flour
Use a heaping tablespoon or medium cookie scoop to drop the cookie dough balls 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.
To make the frosting, beat together the softened butter and cream cheese in a large bowl until smooth and all of the lumps are gone. Sift in the powdered sugar then add the vanilla. Beat until smooth and creamy, about 2 minutes. Frost cooled cookies with cream cheese frosting.
1/2 cup unsalted butter, 8 ounces cream cheese, 3 cups powdered sugar, 1 teaspoon vanilla extract