In a large bowl, use an electric mixer to beat together the butter, oil, sugar, and powdered sugar. Add in the egg and vanilla extract; beat until just combined.
In a separate bowl, whisk together the flour, baking soda, and cream of tartar.
Add the dry ingredients to the wet ingredients and mix until combined.
Cover the dough in plastic wrap and refrigerate for at least an hour (overnight works too).
Preheat oven to 350 F and line 2 cookie sheets with parchment paper or baking mats.
Pour sprinkles into a small plate or shallow bowl.
Roll dough into balls (about 2 tbsp each), roll in sprinkles, then set on prepared baking sheets with 1 inch between each cookie.
Bake for 8-10 minutes, until edges are just beginning to brown.
Allow the cookies to cool for 2 minutes on the cookie sheets before transferring to a wire rack to cool fully.
Notes
While I recommend using cream of tartar, you can use baking powder instead if necessary. Your cookies will have a firmer texture but will still be delicious.