Preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper.
Whisk together the flour, baking soda, salt and nutmeg.
In a stand mixer fitted with the paddle attachment or using an electric hand mixer, cream the butter, sugar and vanilla together on medium-high speed for 3-5 minutes until light and fluffy. Add the eggs one at a time, and scrape down the side of the bowl.
Alternate between the flour mixture and eggnog, adding until all of the ingredients are incorporated. Scrape down the bowl again to make sure everything is evenly combined.
Using a 2-tablespoon scoop, portion the cookie dough onto the prepared baking sheets. These don't spread as much as some cookies, but leave at least 1 1/2 inches in between each cookie (you can always bake in batches to be safe). Allow to chill for 15 minutes.
Bake the cookies for 12-15 minutes, until they're just barely starting to turn golden around the edges. Allow to cool for 5 minutes, then transfer to a cooling rack.
Prepare the frosting
In a stand mixer with the paddle attachment or using an electric hand mixer, beat the butter until light and fluffy. Slowly add the sugar (about 1/4 cup at a time), then add the eggnog (and rum, if using).
Once the cookies are at room temperature, frost, and then top with a generous amount of freshly grated nutmeg (I recommend using a microplane zester for this).
Store cookies in an airtight container, refrigerated, for up to 2 weeks (possibly longer, but they'll taste best within 2 weeks).