Preheat your oven to 350℉. Line your baking sheet with parchment paper.
In a mixing bowl, whisk together the flour, baking powder, baking soda, and pumpkin pie spice. Set aside.
Cream together the butter and sugars until combined and fluffy.
Add the pumpkin, egg, and vanilla to the sugar/butter, and mix until combined.
Add in the flour mixture and keep mixing until it just comes together.
Using a cookie scoop, scoop dollops of dough onto your baking sheet. Lightly flatten them with a finger; it's sticky, so be careful. The cookies won't spread as they bake, but they will rise, so keep that in mind.
Bake for 15 minutes, then transfer to a cooling rack to cool before icing.
To make the icing, beat together the cream cheese, butter, vanilla, and sugar until smooth. Add a dash of milk if it's too thick for your preferences.
Spread the icing over top of your cooled cookies. Top with a dash of pumpkin pie spice and, optionally, some chopped nuts of your choice. I like pecans for the crunch!