Line a sheet tray with parchment paper, set aside. In a medium-sized bowl stir together the flour, baking soda, baking powder, ¼ teaspoon nutmeg, and ¼ teaspoons salt, set aside.
In the body of a stand mixer with the paddle attachment, mix the 1 cup butter until smooth. Add the 1 cup 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 (medium) 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/190°C. 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 ½ cup butter in a large bowl and whip it with an electric hand mixer until smooth. Add 2 cups powdered sugar a little at a time along with the pinch of salt until it’s mixed in. Add the 2 tablespoons eggnog and ½ teaspoon 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
To Store: These cookies can be stored in an airtight container at room temperature for up to 3 days. Cookies can be frozen either frosted or unfrosted for 4-5 months.
Can't find Eggnog? No worries. Replace the eggnog with either half and half or buttermilk in the cookie dough recipe, and use milk or cream in the icing. Add a pinch more nutmeg to replace the lost flavor.
Be Patient: This recipe has a couple of steps in it that might be challenging if you're an inpatient baker. Yes, the dough needs to chill overnight before you bake it. And yes, the cookies need to be left on the sheet trays to cool completely. Skipping either of these steps will likely result in crumbled cookies
Instead of Nutmeg: You can use cinnamon in place of nutmeg in these cookies if you don't really enjoy the flavor of nutmeg. They won't be quite as eggnog-y, but they will still be delicious.