Preheat the oven to 350°F and line 3 baking sheets with parchment paper. Using a stand mixer fitted with the paddle attachment (or a large mixing bowl with a hand mixer), combine the butter, brown and granulated sugar, egg and vanilla and beat on medium speed until creamy and free of lumps. Mix in the pumpkin puree.
In a separate bowl, combine the flour, baking powder, baking soda, pumpkin pie spice and salt. Mix well. Gradually add to the butter mixture and beat on medium speed until just combined. Using a medium cookie scoop, drop the dough by 1 1/2 tablespoons 2 inches apart onto the prepared baking sheets. Bake in the preheated oven for 10 to 12 minutes, or until the bottom edges are golden brown. Remove from the oven, transfer the cookies to a wire rack and let cool.
Using a stand mixer fitted with the whisk attachment (or a large mixing bowl with a hand mixer), combine the confectioners’ sugar, cream cheese, butter and vanilla. Beat on medium speed for 3 minutes.
Spread the frosting on the cookies. Sprinkle with pumpkin pie spice.