Preheat the oven to 350°F. Line a large cookie sheet with parchment paper and set aside.
In a large bowl of a stand mixer fitted with the paddle attachment or a hand mixer, beat the butter and sugar together on medium-high speed for 2 minutes.
1 cup (2 sticks) unsalted butter, 1 cup granulated sugar
Add the egg, vanilla and rum extract then beat well, until the yellow streaks are gone.
1 egg, 1 teaspoon vanilla extract, 3/4 teaspoon rum extract
Add the eggnog and beat again until it's incorporated.
1/2 cup full fat eggnog
Whisk the flour, baking powder, baking soda, salt, and nutmeg in a separate medium bowl. With the mixer on low, add half of the dry ingredients, then turn the speed up to medium to incorporate it fully. Repeat the process with the second half of the dry ingredients. The cookie dough will be wet, this is what you want!
3/4 teaspoon ground nutmeg, 1 teaspoon baking powder, ½ teaspoon salt, 2 1/4 cups all-purpose flour, 1 teaspoon baking soda
Use a heaping tablespoon or medium cookie scoop to drop the dough a few inches apart on a parchment lined baking sheet. Bake for 10 to 12 minutes or until the tops are set and no longer look wet. The cookies will spread out first then start to rise.
Allow to cool on cookie sheet for a few minutes then transfer to wire rack to cool completely before frosting.