In a large bowl, with a mixer on medium speed, beat butter, granulated sugar, and powdered sugar until smooth. Add oil, egg, and vanilla and beat until well blended.
In a medium bowl, mix flour, baking soda, nutmeg, and salt. Stir into butter mixture, then beat until well blended. Cover bowl with plastic wrap and freeze until dough is firm, about 1 hour.
Shape dough into 1-inch balls and place about 2 inches apart on buttered or cooking parchment-lined 12- by 15-inch baking sheets. Flatten each cookie slightly with a floured fork.
Bake cookies in a 350° oven until edges are lightly browned, 8 to 10 minutes; if baking more than one pan at a time, switch pan positions halfway through baking. Let cookies cool on sheets for 5 minutes, then use a wide spatula to transfer to racks to cool completely.
Note: Nutritional analysis is per cookie.
* Percent Daily Values are based on a 2,000 calorie diet. Your daily value may be higher or lower depending on your calorie needs.