In a large mixing bowl, cream together the softened butter, white sugar, and brown sugar until smooth and well combined.
Add the pumpkin puree, egg, and vanilla extract, beating until thoroughly blended.
In another bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, cloves, and pumpkin pie spice.
Gradually add the dry ingredients to the pumpkin mixture, beating until just combined.
Drop tablespoonfuls of dough onto ungreased cookie sheets and flatten slightly with a spoon.
Bake at 350°F (175°C) for 10-12 minutes, or until the edges are lightly browned.
Remove from the oven and transfer the cookies to a wire rack to cool completely.
In a medium bowl, place the powdered sugar and set aside.
Melt the unsalted butter in a small saucepan over low to medium heat, stirring frequently.
Cook until the butter turns a deep golden brown with a nutty aroma, about 10 minutes. Remove from heat and let cool for 2-3 minutes.
Carefully pour the brown butter over the powdered sugar. Add milk and vanilla, whisking vigorously until smooth. Adjust the consistency with more milk or powdered sugar if needed.
Spread about 1 tablespoon of icing onto each cooled cookie. Allow the icing to set before serving.