1. Weigh or lightly spoon flour into dry measuring cups; level with a knife. Combine flour, cinnamon, nutmeg, baking soda, and salt in a medium bowl, stirring with a whisk. Combine sugar, butter, and oil in a large bowl; beat with a mixer at medium speed until smooth. Add syrup, vanilla, and egg white to sugar mixture, beating until well blended. Gradually add flour mixture to sugar mixture; beat at low speed just until combined. Divide dough in half; roll each half into a ball. Cover with plastic wrap; chill 1 hour.
2. Working with 1 portion at a time, gently press dough into a 4-inch square on heavy-duty plastic wrap. Cover with additional plastic wrap. Roll dough, still covered, to a 1/16-inch thickness. Repeat procedure with the remaining dough; freeze 30 minutes.
3. Preheat oven to 375°.
4. Working with 1 portion at a time, remove top sheet of plastic wrap. Turn dough over; remove remaining plastic wrap. Cut with a 2-inch round cutter into 22 cookies; place cookies 2 inches apart on a baking sheet lined with parchment paper. Repeat procedure with remaining dough portion. Bake at 375° for 8 minutes or until lightly browned. Cool on pans 5 minutes. Remove from pans; cool on wire racks. Repeat procedure with remaining dough.