Preheat oven to 350 degrees F. Line two baking sheets with parchment paper. Set aside.
In a large mixing bowl, beat together the butter, sour cream, and sugar for 2 minutes until fluffy. Mix in the eggs and vanilla until smooth.
In a medium mixing bowl, combine flour, baking soda, salt, and cream of tartar. Add the dry ingredients to the wet ingredients and mix until just combined. Scrape the side of the bowl as needed.
Using a cookie scoop, portion the dough onto the prepared baking sheets 2 inches apart. With floured hands, roll each scoop into a ball. Use the bottom of a drinking glass dipped into sugar to flatten the cookies.
Bake for 10 minutes. Do not overbake! Remove from the oven and let the cookies rest on the baking sheet for 2 minutes before transferring them to a wire rack to cool completely. Repeat with the remaining cookie dough.
Frosting
Beat the butter and shortening together in a large mixing bowl, and then mix in the pumpkin puree. Add the powdered sugar to the bowl, one cup at a time, mixing after each addition. Mix in the pumpkin pie spice and salt. If frosting is too thick, add a little milk or water. If the frosting is too thin, add more powdered sugar as needed until your desired consistency is reached.
Transfer the frosting to a piping bag fitted with the piping tip of your choice. Frosting the cookies once they're completely cooled.