In a large bowl, use an electric mixer to combine the butter with the white and brown sugars; mix until well combined. Add the pumpkin puree, egg and vanilla and mix until incorporated.
In a medium bowl, whisk together the flour, cream of tartar, cinnamon, baking soda, salt, ginger, nutmeg and allspice. Add the dry ingredients to the wet mixture, and mix just until incorporated.
Cover the bowl with plastic wrap and refrigerate until slightly firm (about 2 hours).
When ready to bake, preheat the oven to 350 degrees F. and line baking sheets with parchment paper. Combine the rolling sugar ingredients in a bowl.
Remove the dough from the refrigerator. Use a medium cookie scoop (1½ tablespoons) or large spoon to scoop out the dough. It will be sticky. It's easiest to drop the dough into the rolling sugar, let it pick up some of the sugar, and then put it into your hands to roll into a ball. Roll it again in the sugar, and then place cookie dough balls on the prepared baking sheet. Repeat with remaining cookie dough, placing the cookie dough balls about 2 inches apart on the baking sheet.
Bake for 13 to 15 minutes, or until the cookies are slightly firm to the touch on top. Let them cool on baking sheets for 5 minutes, and then transfer them to a rack to cool completely.
Notes
Store these cookies in an airtight container. They freeze well too.