Preheat oven to 350°F. Line baking sheets with parchment paper. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, combine butter, granulated sugar, and confectioners' sugar. Mix on medium speed until just combined.
Add oil and add eggs. Mix well.
In a separate bowl, whisk together the flour, pumpkin pie spice, cream of tartar, and salt.
Slowly add the flour mixture to the butter mixture. Continue mixing until the dough pulls away from the sides of the bowl.
Using a 2-tablespoon scoop, drop dough onto a lined baking sheet.
Roll each cookie dough ball until it is smooth and round. Set each cookie dough ball back onto a lined baking sheet, about 2 inches apart. (I usually put 6 cookie dough balls onto each cookie sheet.)
Using the bottom of a wet kitchen glass that has been pressed into granulated sugar, press down on each cookie dough ball until the center of the cookie is pressed flatter than the edges (like a well in the center). Repeat this on all cookies.
Bake for 7-9 minutes, or until edges are just slightly starting to brown and the center no longer appears wet. Allow the cookies to cool as you make the frosting.
NOTE: It is important to NOT over-bake the cookies. Be sure to check your cookies at 7 minutes. Every oven is different and overbaking can result in a tough, crumbly, tasteless cookie.