Preheat oven to 350°F and line a baking sheet with parchment paper.
In a mixer, beat the sugar, brown sugar, and butter for 3 minutes until fluffy.
Add eggs one at a time, mixing well, then stir in the vanilla and pumpkin puree.
In a separate bowl, whisk together flour, baking soda, spices, salt, and baking powder.
Gradually add the dry ingredients to the wet mixture, stirring just until combined.
Scoop 1½-tablespoon portions of dough onto the baking sheet, spacing 2 inches apart.
Bake for 18 minutes, then transfer cookies to a cooling rack.
For the icing, whisk together powdered sugar, cinnamon, and milk until thick; spoon over cooled cookies and let set for 1 hour.