Preheat oven to 350 degrees F.
In the bowl of a stand mixer beat butter on medium speed for 30 seconds. Add granulated sugar, baking powder, baking soda, salt, cinnamon, pumpkin pie spice, cloves and nutmeg. Beat until combined.
Beat in the eggs and vanilla until combined. Then, add the pumpkin. Beat in as much of the flour as you can using the mixer. Stir in remaining flour with a wooden spoon.
Drop dough by heaping teaspoons 2 inches apart on ungreased cookie sheets. Bake for 10 to 12 minutes or until tops are set. Transfer to wire racks to cool.
For frosting, in a small saucepan heat butter and brown sugar until melted and smooth. Transfer to a medium bowl. Stir in milk and vanilla. Beat in powdered sugar until smooth. (I made a double batch of the frosting and ended up using all of it. I recommend doing the same.)
Spread frosting on the cooled cookies. Sprinkle with additional cinnamon, if desired.