Add eggnog, eggs, and vanilla and mix until combined.
Whisk flour, baking powder, baking soda, salt, and nutmeg in a separate bowl.
Add dry ingredients to wet ingredients and mix just until combined.
Chill dough at least 1 hour.
Scoop dough onto parchment-lined baking sheets.
Bake until cookies are set.
Flatten cookies gently with a greased glass right after baking if desired.
Cool completely.
Mix frosting ingredients until smooth, frost cooled cookies, and let frosting set.
Notes
Chill the dough so the cookies do not spread too much. Do not frost warm cookies or the icing will slide. Grease the bottom of the glass when flattening so it does not stick. Store once frosting is fully set so cookies can be stacked for gifting.