Preheat the oven to 350 degrees F. Line baking sheets with parchment paper.
Cream together the butter and sugar in a large mixing bowl with an electric mixer, or in a stand mixer with a beater blade.
Add the eggs and vanilla. Whisk the mixture again until smooth and homogenous.
Gently fold in the pumpkin puree until it is fully incorporated with the other wet ingredients.
Add the baking powder, baking soda, pumpkin spice, and flour to the bowl. Beat the dry ingredients just combined and leave in the bowl. Place the cookie dough in the fridge for 30 minutes.
Make the frosting by whisking together the cream cheese, butter, vanilla extract, and powdered sugar until smooth and homogenous.
Drop 1.5-inch sugar cookie dough balls on the prepared baking sheets. Bake for 12 minutes or until edges are set and cookies are done in the middle. Remove cookies and move to a rack to cool completely.
Allow the cookies to cool slightly before frosting. Use a knife to apply frosting to the tops of all the cookies. Let set and serve.
Notes
It is important to let the cookies cool a little bit before you frost them, otherwise the frosting will run off the cookies.
You can press down gently on the cookies as soon as they come out of the oven to make a small flat spot that gives you a nice spot for the frosting.