Preheat your oven to 350°F (175°C) and line two baking sheets with parchment paper or silicone mats.
In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, cloves, and salt.
In a large mixing bowl, cream the softened butter and granulated sugar until light and fluffy, about 2-3 minutes.
Add the pumpkin puree, egg, and vanilla extract to the butter-sugar mixture and beat until smooth.
Gradually mix in the dry ingredients, stirring until just combined. Be careful not to overmix.
Using a tablespoon or cookie scoop, drop the dough onto the prepared baking sheets, spacing them 2 inches apart.
Bake for 12-15 minutes, or until the edges are set. Let cool on the baking sheets for 5 minutes before transferring to a wire rack.
For the icing, whisk together the powdered sugar, melted butter, vanilla extract, and milk until smooth.
Once the cookies are completely cool, drizzle the icing over the tops using a spoon or piping bag. Let the icing set for 20 minutes before serving.