In a medium bowl, combine flours, baking powder, and salt.
In large bowl, combine sugar, oil, and butter; beat with a stand or hand mixer at low speed until blended (about 3 minutes). Increase speed to high; beat until light and creamy (about 3 minutes), occasionally scraping side of bowl with rubber spatula. Reduce speed to low; add egg whites and vanilla, beating until smooth. Add flour mixture to sugar mixture; beat just until blended.
Divide dough in half; flatten each half into a disk. Wrap each disk with plastic wrap; refrigerate 2 hours or until dough is firm enough to roll.
Preheat oven to 375°. Using a floured rolling pin, roll 1 dough disc to 1/8 inch thick on a lightly floured surface. Use cookie cutters to cut out as many cookies as possible. With a lightly floured spatula, place cookies 1 inch apart on an ungreased large baking sheet.
Bake cookies until lightly browned (about 10-12 minutes). Transfer cookies to wire rack; cool. Top with meringue icing. Repeat procedure with remaining dough and trimmings.