Preheat oven to 350 degrees F. In the bowl of a stand mixer, beat butter on medium speed for 2-3 minutes. 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 spatula.
Drop dough by heaping teaspoons 2 inches apart on ungreased cookie sheets. Bake the cookies for 10-12 minutes or until the tops are set. Transfer the cookies to wire racks to cool.
For frosting: In a small saucepan, heat butter, brown sugar, and instant coffee until melted and smooth. Transfer to a medium bowl. Stir in milk and vanilla. Beat in powdered sugar until smooth.
Spread frosting on the cooled cookies. Sprinkle with additional cinnamon, if desired.