Preheat oven to 350°F. Line two baking sheets with parchment.
Whisk flour, baking soda, cream of tartar, and salt in a medium bowl; set aside.
Beat butter, oil, granulated sugar, and powdered sugar until light and creamy, 2–3 minutes.
Add eggs and vanilla; beat again until smooth and well combined.
Stir in the dry ingredients in two additions, mixing just until a soft dough forms.
Scoop dough into 1½-tablespoon portions. Roll each ball in granulated sugar if using.
Arrange on sheets 2 inches apart. Gently flatten tops with the bottom of a glass.
Bake 8–10 minutes, until edges look set and centers appear soft but not wet.
Cool on the sheet for 5 minutes, then transfer to a rack to finish cooling.