In the bowl of an electric mixer, beat together the butter and sugars on medium-high speed until light and fluffy, about 2 to 3 minutes. Stir in the pumpkin puree, then beat in the egg and vanilla.
In a separate medium bowl, whisk together the flour, baking powder, salt, cinnamon and nutmeg. With the mixer on low speed, gradually add the dry ingredients to the wet ingredients. Stir until fully incorporated. Cover and chill the dough for at least an hour.
Preheat the oven to 350 degrees. Line baking sheets with parchment paper or silicone liner. Mix the coating ingredients in a small bowl. Scoop equal dough balls and roll into the sugar mixture. Set on cookie sheet, about 2 inches apart. Dip the bottom of a flat drinking glass into water, then into the sugar mixture and use to slightly flatten the dough balls. Recoat the bottom of the glass in the sugar-spice mixture as needed.
Bake the cookies for 12 to 14 minutes, or until baked through. Let cool on the baking sheets about 5 minutes, then transfer to a wire rack to cool completely. Repeat with remaining dough.