In a large mixing bowl, cream together the butter, granulated sugar, and powdered sugar until light and fluffy.
Beat in the eggs, one at a time, followed by the vegetable oil and vanilla extract.
In a separate bowl, whisk together the flour, baking soda, cream of tartar, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Cover the dough and refrigerate for at least 2 hours or overnight.
Preheat the oven to 375°F (190°C) and line baking sheets with parchment paper.
Roll the chilled dough into 1-inch balls and place them on the prepared baking sheets, spacing them about 2 inches apart.
Use the bottom of a glass or a cookie stamp to gently flatten each cookie.
Bake for 10-12 minutes, or until the edges are lightly golden.
Allow the cookies to cool on the baking sheets for a few minutes before transferring them to wire racks to cool completely.
Store in an airtight container at room temperature.