Preheat the oven to 350ºF (180ºC) and line two baking sheets with parchment paper.
Whisk the flour, cinnamon, nutmeg, baking powder, baking soda, and salt in a large mixing bowl until well combined. Set aside.
Place the butter and both sugars in the bowl of a stand mixer. Using the whisk attachment, cream the ingredients together on medium-high speed until fluffy, about 1 to 2 minutes. Stop the mixer and scrape down the sides of the bowl halfway through mixing.
Add the egg and rum extract. Mix until well combined. Stop the mixer and scrape down the sides of the bowl, then add the eggnog and mix again until combined. At this point the mixture may look a bit curdled. This is normal, especially if your eggnog is a bit cold from the fridge and has cooled the butter.
Place a fine mesh sieve over the mixing bowl. Scoop the flour mixture into the sieve. Use the back of a spoon to press and sift the flour into the bowl of wet ingredients. Some of the ground nutmeg may be too large to pass through the sieve so just dump it into the bowl after all of the flour is sifted in.
Switch to the paddle attachment. Mix the cookie dough on low until the flour is justcombined. Remove the bowl from the mixer and use a rubber spatula or wooden spoon to finish mixing the dough by hand. Be sure to stir underneath all of the dough to incorporate any loose flour.
Use a small cookie scoop (about 2 teaspoons) to scoop 12 portions of dough onto each prepared baking sheet, spaced about 2 inches apart. Note that the dough is very soft! For best results, fill the scoop with dough then press it firmly along the inside of the bowl to level it off, removing any excess dough from the scoop.
Place the cookies in the preheated oven. Bake them for 12 to 14 minutes, or until the bottoms are light gold. Cutting a cookie in half is the best way to be sure they are baked through. Remove the cookies from the oven and cool them on the baking sheet for 10 minutes before transferring them to a wire rack to cool completely before frosting.