In an electric mixer or large bowl using a hand mixer, beat together the oil, butter, and sugar. Whip until smooth about 1 minute.
Mix in the eggs and vanilla. Add flour, cream of tartar, baking soda, and salt. Mix until well blended.
Place the batter in the refrigerator for at least 1 hour to firm up.
Preheat oven to 350 degrees. Line 2 baking pans (cooking sheets) with parchment paper, see shopping tip.
In a small bowl, add the ⅓ cup sugar for rolling.
Scoop the batter into rounded 1-inch pieces and roll in sugar. Place each on prepared baking pans.
Bake the cookies for 11 minutes, then rotate pans and bake for about 12 more minutes. This will create a crispy cookie when cooled. If you desire softer cookies, bake for 7-8 minutes, rotate pans, and bake for another 7-8 minutes. Repeat with remaining batter.
Store cooled cookies in an airtight container. Crispy cookies will last for several weeks. Soft ones will last a few days if they last that long!