Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
In a medium bowl, whisk together flour, baking powder, baking soda, salt, cinnamon, nutmeg, ginger, and cloves.
In a large mixing bowl, beat softened butter with granulated sugar and brown sugar until light and fluffy.
Add egg, vanilla, and pumpkin puree. Mix until combined.
Gradually add dry ingredients to wet ingredients, mixing until a soft dough forms.
Scoop tablespoon-sized portions of dough onto prepared baking sheets, spacing 2 inches apart.
Bake 10–12 minutes until cookies are set and lightly golden on edges. Cool completely on wire racks.
Meanwhile, make the frosting: beat softened cream cheese and butter until smooth. Add powdered sugar and vanilla, mixing until fluffy. Add milk, 1 tablespoon at a time, until desired consistency is reached.
Frost cooled cookies generously with cream cheese frosting.
Optional: sprinkle with cinnamon sugar or decorate with festive sprinkles.