Spread pumpkin puree onto a paper-towel-lined plate. Cover with more paper towels, press firmly, and blot for 5 minutes until no more liquid seeps into the towels.
Whisk flour, baking soda, baking powder, salt, cinnamon, nutmeg, ginger, and cloves in a medium bowl until evenly combined. Set aside.
Beat softened butter, granulated sugar, and brown sugar on medium speed for 3 minutes until pale and fluffy.
Add egg and vanilla extract. Mix on medium speed for 1 minute until smooth. Scrape down the bowl.
Add blotted pumpkin puree. Mix on low for 30 seconds until no orange streaks remain.
Add the flour mixture in two additions, mixing on low after each just until no dry streaks remain. Do not overmix.
Cover the bowl and refrigerate for 30 minutes.
Preheat oven to 175°C (350°F). Line two baking sheets with parchment paper.
Scoop 1.5-tablespoon portions onto prepared sheets, spacing 5 cm (2 inches) apart. Bake one sheet at a time on the middle rack for 12 to 13 minutes, until edges are set and matte. Centers should look slightly underdone.
Rest on the pan for 5 minutes, then transfer to a wire rack. Cool completely, at least 20 minutes, before glazing.
Beat cream cheese until smooth. Add sifted powdered sugar and vanilla; mix until combined. Add milk one tablespoon at a time until the glaze drizzles slowly off a spoon. Drizzle or spread over cooled cookies.