Preheat the oven to 375 degrees F (190 degrees C). Grease cookie sheets.
In a large bowl, mix together the butter, oil, white sugar, and confectioners' sugar until smooth. Beat in the eggs one at a time, then stir in the vanilla. Combine the flour, baking soda, and cream of tartar. Stir into the sugar mixture until just combined. Drop dough by teaspoonfuls onto the prepared cookie sheets.
Bake for 8 to 10 minutes in preheated oven, or until bottoms are lightly browned. Remove from baking sheets to cool on wire racks.