In a large mixing bowl, beat the butter until creamy. Add the vegetable oil, granulated sugar, and powdered sugar, beating until well combined.
Mix in the egg and vanilla extract until fully incorporated.
Cover the dough and refrigerate for at least 1 hour to firm up.
Mix 1 cup powdered sugar with nutmeg, cinnamon, and enough eggnog to achieve a spreadable consistency. Drizzle over cooled cookies.