Spread the pumpkin puree on paper towels and press with more towels on top to blot out as much water as possible.
Beat the butter, granulated sugar, and brown sugar together until light and fluffy, about 2 minutes.
Mix in the blotted pumpkin, molasses, egg, and vanilla until smooth. It may look slightly curdled, which is fine.
In a separate bowl, whisk the flour, baking soda, cinnamon, ginger, cloves, and salt. Add to the wet ingredients in two or three additions, stirring just until combined.
Cover and chill the dough for at least 1 hour.
Roll and Bake
Preheat the oven to 350F and line two baking sheets with parchment.
Scoop tablespoon-sized balls, roll each in the reserved granulated sugar, and space them 2 inches apart on the sheets.
Bake 10 to 12 minutes, until the tops are cracked and the edges look set.
Let the cookies rest on the sheet for 3 minutes, then transfer to a wire rack to cool completely.
Notes
Blot the pumpkin: This is the key to chewy, not cakey, cookies.
Don't overbake: Pull them when the edges set but the centers still look soft.
Make ahead: Freeze the sugared dough balls and bake from frozen, adding a minute or two.