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 puree and beat until incorporated. With the mixer running on low, gradually add the flour, then increase the speed to medium and mix just until combined. The dough should be soft and sticky, similar to a thick drop-cookie batter.
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 portions of dough a few inches apart on the prepared baking sheet. The cookies will puff and remain fairly thick rather than spreading significantly.
Bake for 10 to 12 minutes, until the tops are set, no longer look wet, and spring back lightly when touched. Do not wait for the tops to become deeply golden.
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