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1. In a large bowl, beat butter and oil with an electric mixer on medium to high speed for 30 seconds. Add 1/2 cup granulated sugar, powdered sugar, baking soda, cream of tartar, and salt. Beat until combined, scraping sides of bowl occasionally. Beat in egg and vanilla until combined. Beat in the flour. Cover and chill for 4 to 24 hours or until dough is easy to handle.
2. Preheat oven to 375 degrees F. Shape dough into 1/2-inch balls for small cookies or 1 1/4-inch balls for larger cookies. Place small balls 1 1/2 inches apart on ungreased cookie sheets or large balls 2 inches apart. Flatten balls with the bottom of a glass dipped in granulated or colored sugar.* Or, if desired, sprinkle flattened cookies with a mixture of granulated sugar and edible glitter.
3. Bake for 8 to 10 minutes for small or 10 to 12 minutes for large cookies or until bottoms are lightly browned. Transfer cookies to wire racks and let cool. Makes 72 small cookies or 36 large cookies