In a stand mixer, combine sugar, powdered sugar, egg, butter, oil, and vanilla. Mix on medium speed until fully encorporated.
In a separate bowl, combine flour, baking soda, salt, and cream of tartar. Slowly add to the stand mixer. Stir until fully incorporated. Scrape down sides as necessary.
Chill the dough in a fridge overnight, or in the freezer for at least an hour. (see note)
Roll dough into small balls (approximately 1-2 Tbsp). Roll in granulated sugar and place on a cookie sheet. Leave 1½-2 inches between cookies as they will spread out considerably. Press cookies until about ½ an inch tall. You can use either a cookie press, a fork, or your palm.
Bake at 350° for 10-15 minutes. Cookies will have spread out thin, and turned lightly tan around the edges.
Notes
Many cookie recipes call for chilling in order to get more even baking and a nicer shape, when that is the case it can usually be skipped if your rushed for time. However this is not one of those recipes. Due to the oil and butter, this recipe requires chilling time to even be able to roll it into balls - so please plan accordingly.
If you wish to save some of this dough for later, after the dough has been rolled into balls, place in a zip lock back and add to freezer. When you're ready to bake, allow them to thaw for 20 minutes, then roll in sugar, press, and bake as normal.