Cream butter, oil, sugar and powdered sugar in a large bowl with an electric mixer. Beat in egg and vanilla extract. Combine flour, baking powder, salt and cream of tartar. Gradually add to creamed mixture and mix well. Chill for 2 to 3 hours.
Roll dough into 1-inch balls. Place on ungreased baking sheet. Dip a flat bottomed drinking glass into sugar and gently flatten each cookie.
Bake at 350°F for 9 to 11 minutes or until edges of cookies are lightly browned. Immediately remove from baking sheets and cool on racks.
Recipe Tip: Use a glass with a molded bottom for a more decorative appearance.