Add the egg, pumpkin spice and vanilla, and beat to incorporate.
Add the baking soda, cream of tartar and salt, and then add the flour in 2 or 3 batches, beating each batch of flour to combine well.
Use a medium cookie scoop or a small ice cream scoop to form dough balls about 2 tablespoons each. Arrange the dough balls on the prepared baking sheets with space in between to allow for spreading.
Bake the cookies for about 10 to 15 minutes, just until the bottoms begin to turn golden brown.
Remove the baking sheet from the oven, and let the cookies cool for about 5 minutes before transferring them to a wire rack to cool completely.
While the cookies cool, make the frosting.
In a medium bowl, beat the butter until smooth and pale. Add the remaining tablespoon of cream if needed.
Add the salt and slowly add the confectioners sugar, beating until the mixture is crumbly.
Add the vanilla, pumpkin pie spice and 2 tablespoons of the cream. Beat until the frosting is smooth.
Spread the frosting on top of the cooled cookies.
Dust with pumpkin spice while the frosting is still wet (optional).
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