Preheat oven to 350 degrees and line a baking sheet with parchment paper.
In a bowl, cream butter, brown sugar, and granulated sugar together until well combined.
Add in the egg, and mix until well combined.
Add in pumpkin puree, mixing until combined.
Next, add in allspice, nutmeg, cloves, cinnamon, vanilla extract, baking powder, baking soda, and salt. Mix until everything is combined.
Add in flour and mix until flour is well incorporated.
With a small-med sized cookie scoop (about 1 1/2 - 2 Tbsp), scoop dough and place evenly on your prepared cookie sheet.
Bake for about 12-13 minutes, or until cookies are set. Allow to cool completely before frosting.
With a standing mixer or hand mixer beat butter until light and fluffy, about 1-2 minutes. Add in maple syrup and vanilla extract. Beat until well combined.
Add in powdered sugar, 1/2 cup at a time, mixing until well combined (beginning with 1 1/2 cups).
Add in 2 tsp of milk and beat until combined.
If frosting feels too runny, add in the remaining 1/2 cup of powdered sugar. If it feels too thick, add in the remaining tsp of milk.
Spread frosting on cookies.
Enjoy!