Preheat the oven to 350 degrees. Line your cookie sheet with parchment, set aside.
For the cookies - In the bowl of a stand mixer, combine the softened butter, sugar and brown sugar. Cream until smooth.
Add in the egg, vanilla and eggnog. Mix thoroughly on medium speed.
Add in all the remaining dry ingredients: flour, cornstarch, baking powder, baking soda, cinnamon and salt. Mix well. Stop the machine, scrape down the sides and mix again.
After mixing the cookie dough, chill it in the refrigerator for at least 30 minutes. This makes the dough easier to handle and prevents the cookies from spreading too much during baking.
Using a heaping tablespoon or a small cookie scoop, scoop out a ball of dough about the size of a ping-pong ball. Place on the parchment lined baking sheets.
Bake in the preheated oven for 10 minutes until golden brown.
Immediately upon removing the baking tray from the oven, use a flat-bottomed drinking cup or similar item to press the cookie into a flat disc. Allow to cool completely before frosting.
To make the eggnog frosting - in the bowl of a stand mixer, combine the softened butter, salt, vanilla, and the powdered sugar. Mix on low until combined and then bump up the speed to medium for another minute.
Add in one tablespoon of eggnog at a time, mixing well between each addition. Mix on medium / high speed for another minute to fluff up the frosting once all the eggnog has been mixed in.
Transfer the frosting to a piping bag fitted with a large star tip.
Start in the center of the cookie, and using steady, even pressure pipe the frosting outward creating a spiral. Repeat with remaining cookies.
Add a sprinkle of nutmeg or cinnamon.