Start by browning the butter. You'll take ½ cup of butter and melt in a medium saucepan, and then continue to cook past the point of boiling. You will do this over medium heat, past the point of the butter foaming, until it turns a nice golden brown, and has a nutty smell. Make sure you are stirring and watching it, because it will turn from perfectly brown to burnt fairly quickly. Once finished browning, pour into a bowl and put back in the fridge to solidify again. You will want it to become a soft solid again to start the cookie dough. This should take about an hour.
Preheat oven to 350 F. Line two baking sheets with parchment paper.
Place browned butter, oil and sugars in the bowl of a stand mixer, fitted with a paddle attachment. Beat on medium-high speed until light and fluffy, 3 to 4 minutes.
Add eggs one at a time, mix until smooth on low speed, about 30 seconds each egg. Beat in the vanilla and pumpkin.
In a separate bowl, whisk together salt, cream of tartar, baking soda, pumpkin pie spice and flour. Add flour mixture to bowl and mix on low, being careful not to overmix. I usually like to stop mixing when I still have flour remnants around the bowl.
Scoop the dough into large balls. I like to use a ¼ cup cookie scoop.
Place 8 cookie balls on your baking sheet. Flatten cookie balls with a glass until they are about ½ inch thick. I like to spray the bottom of my glass with cooking spray and dip in sugar before flattening. Bake the cookies for 10-11 minutes. You're looking for a light golden brown outside rim. Allow to cool completely before frosting.