Preheat oven to 350ºF. Line a baking sheet with parchment paper.
In a large mixing bowl or stand mixer cream together the butter, sugar and brown sugar. Add in the egg, vanilla and eggnog. Continue to mix until well combined and a soft, creamy mixture is formed.
In a separate mixing bowl whisk together the flour, baking soda, salt, cinnamon and nutmeg until well combined.
Add the flour mixture to the butter mixture and mix until a soft dough forms.
Using a cookie scoop or spoon, drop the cookies onto the baking sheet. With the back of a drinking cup, press the cookies until they are 1/4 inch thick. If the cup sticks to the cookies, grease with a little oil or butter.
Bake the cookies for 8-10 minutes, until golden brown around the edges. Place on a cooling rack to cool.
How to Make Eggnog Buttercream Frosting
In a mixing bowl with an electric mixer combine the butter, powdered sugar, Eggnog and vanilla. Mix until soft frosting has formed. Frost the cooled cookies.
Mix the nutmeg and cinnamon, sprinkle over the top of the frosted cookies.