Preheat oven to 375 degrees.
Combine the flour, baking powder, baking soda, salt, cinnamon, ginger, nutmeg, and clove in a bowl until mixed thoroughly set aside. Using a mixer set on medium speed, whip together butter, brown sugar, and white sugar until pale and fluffy, about 3 - 4 minutes. Add in egg, mixing after the addition until combined. Mix in the sour cream, vanilla extract, and pumpkin puree then mix well. Slowly add dry ingredients and mix until combined.
Transfer the cookie dough to a zip lock bag then cut the tip and pipe 1 1/2-inch rounds onto a nonstick or Silpat lined cookie sheet, spacing cookies at least 1 inch apart. Bake in the oven for about 12 minutes until tops spring back when touched. Allow to cool several minutes on baking sheet before transferring to a wire rack.
Prepare the browned butter icing. Measure the powdered sugar into a mixing bowl, set aside. Brown butter in a small saucepan over medium heat, swirling pan occasionally. Remove from heat when the golden brown flecks begin to appear in the center of the foamy bubbles. Carefully pour browned butter over powdered sugar in mixing bowl, use a rubber spatula to scrape any excess butter and browned bits from pan. Add the half and half and stir mixture until smooth and creamy. Spread over cookies with a spoon. Add extra half and half, about 1 - 2 tsp at a time, to icing mixture to thin because it will thicken as it sits. Side Note: The icing is very sweet so don't put it on too thick.
Store in an airtight container in a single layer in refrigerator and rest at room temperature before serving.