Preheat oven to 350 F. Line a baking sheet with parchment paper or a silicone baking mat.
In a large bowl beat the butter on medium speed for 30 seconds.
Add sugar, baking powder, baking soda, salt, and pumpkin pie spice. Beat until combined.
Beat in the egg and vanilla until combined.
Mix in pumpkin.
Stir in flour until incorporated.
Drop 1.5 tablespoons of dough 2 inches apart on prepared baking sheet.
Bake for 10-12 minutes or until tops are set.
Transfer to wire racks to cool.
For frosting, beat butter until smooth.
Add 2 tablespoons milk and vanilla extract. Beat until combined.
Continue beating, and add powdered sugar 1/2 cup at a time.
Add in additional milk to make desired consistency.
Spread frosting on the tops of the cooled cookies.
Sprinkle with additional cinnamon if desired.
Notes
Butter: I use salted butter in both the cookies and the frosting. If using unsalted butter, increase the salt in the cookies to 3/4 teaspoon and add a couple of pinches of salt to the frosting.
Pumpkin pie spice: Store-bought or make your own with this recipe.
Pumpkin puree: Be sure to use pumpkin puree, not pumpkin pie mix.
Flour: To properly measure the flour, either weight it or sift/stir to break it up, lightly spoon into the measuring cup, and level.
Milk: Any milk, from skim to whole, will work. Cream will also work.
Powdered sugar: If your powdered sugar is lumpy, sift to break it up before using. This will help it to incorporate into the frosting better.