Position oven racks to upper and lower thirds of the oven. Preheat the oven to 350°. Line two baking sheets with parchment paper and set aside.
Use a mixer to cream the sugar and butter together until fluffy. Add the egg and mix well.
In a medium bowl, whisk together the flour baking soda, cream of tartar and salt. Add the dry ingredients to the butter mixture and blend until a soft dough forms.
Make the pumpkin spice
In a small bowl combine the sugar, cinnamon, ginger, nutmeg, allspice and cloves and mix well to combine.
Scoop out dough a rounded teaspoon at a time and roll into 1" balls. Dunk the cookie dough into the spice mix and coat completely.
Transfer the cookie to the baking sheet and lightly tamp down on the dough. Continue the process for the remaining cookies giving them a few inches of space between - as the cookies will spread to about 2½".
Place one cookie sheet on the top rack and the other on the bottom. Bake for 8 minutes, then swap the cookie sheets so the bottom sheet moves to the top rack and vice versa.
Bake for an additional 8 minutes. Remove from oven and cool on the cookie sheet for 5 minutes before transferring to a wire rack to cool completely.