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Beat butter and oil in a large bowl with an electric mixer on medium to high speed for 30 seconds. Add 1/2 cup granulated sugar, confectioners’ 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.
Preheat oven to 375°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 ( or large balls 2 inches apart)on ungreased baking sheets. 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, colored sugar and/or edible glitter.
Bake for 8 to 10 minutes for small cookies (or 10 to 12 minutes for large) or until bottoms are lightly browned. Transfer cookies to wire racks and let cool.