Line a sheet tray with parchment paper, set aside. In a medium-sized bowl stir together the flour, baking soda, baking powder, nutmeg, and salt, set aside.
In the body of a stand mixer with the paddle attachment, mix the butter until smooth. Add the powdered sugar and brown sugar, mix until combined. Add the egg and mix it in until combined. Add the flour and stir it in. Add the eggnog, rum extract, and vanilla extract, stir it in until combined, scrape the sides as needed.
Place 1.5 tablespoon-sized dollops of the dough on the sheet tray close, but not touching. I like to use a 1.5 tablespoon cookie scoop for this. Cover with plastic wrap and place in the fridge for at least 4 hours, best overnight.
Preheat the oven to 375°F. Line sheet trays with parchment paper, set aside.
Roll the dough balls into smooth balls and place them on the prepared sheet trays 2 inches apart. Gently press down on the tops with the heel of your hand. Bake for 8-9 minutes until the bottom edges are lightly browned and the tops do not appear glossy.
Let the cookies cool on the sheet tray completely.
While the cookies cool, make the frosting. Place the butter in a large bowl and whip it with an electric hand mixer until smooth. Add the powdered sugar a little at a time along with the pinch of salt until it’s mixed in. Add the eggnog and rum extract. Mix it in until combined, turn up the speed to medium-high and whip the frosting for 3 minutes until light and fluffy.
Use a butter knife to frost the tops of the cookies, using about a teaspoon of frosting per cookie. Add a garnish of more ground nutmeg on top if desired.
Notes
Stays fresh in an airtight container on the counter for up to 3 days. You can freeze them frosted or unfrosted for 4-5 months.