Begin by making the cookies: Preheat the oven to 375 degrees. Line two baking sheets with parchment paper.
Sift together the flour, baking powder, baking soda, salt, and pumpkin spice into a bowl. Set the bowl aside.
In the bowl of an electric mixer with the paddle attachment on, cream together the butter and sugar. Beat them for four minutes, scraping down sides of the bowl.
Add the egg, pumpkin, and vanilla extract and beat them together for one minute.
Add the dry ingredients and mix them up until no streaks for flour remain.
Drop two tablespoons of dough per cookie on to the baking sheets. Space the cookies one inch apart. Bake one pan at a time for 12 minutes, until the edges start to brown. Cool the cookies on the pans for five minutes and then on wire racks until they're completely cool.
While the cookies bake, make the frosting: In the bowl of an electric mixer with the paddle attachment, cream together all the ingredients. Spread a generous amount on each cooled cookie.