In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt; set aside. In a large bowl, use an electric mixer to blend together the butter, brown sugar, and granulated sugar until light and smooth, about 1 minute. Blend in the eggnog, egg, vanilla, and rum extract until well-combined. Gradually mix in the flour mixture until fully-incorporated, scraping down the sides of the bowl halfway through. Tightly cover the bowl with plastic wrap and refrigerate for at least 2 hours.
Place rack in center position of oven. Preheat oven to 350°F. Line a rimmed sheet pan with parchment paper.
Remove the cookie dough from the refrigerator and scoop into balls (you can use a tablespoon or a 1.5-inch cookie scoop). Arrange the cookies 2 inches apart on the prepared baking sheet. Bake for 11 to 13 minutes or until the cookies are puffed up and set in the center. Remove the pan from the oven and allow the cookies to cool for 10 minutes before transferring them to a rack to finish cooling.
Once the cookies are cool, measure the powdered sugar, 2 tablespoons eggnog, vanilla, and rum extract into a medium bowl. Whisk together until combined, adding additional eggnog one teaspoon at a time until the glaze is just thin enough to drizzle. Glaze the cookies and decorate with sprinkles as desired. Allow the glaze to set completely (for several hours) before storing in an airtight container.
All ingredients should be at room temperature. You don't want your creamed butter to lump back up due to adding cold eggnog!
Blend the flour mixture into the wet mixture thoroughly, or streaks of eggnog may "run" out of the cookies as they bake.
The rum extract is optional.
This dough will be too sticky to work with right after mixing it together. Scrape down the sides of the bowl, cover with plastic wrap, and chill the dough in the refrigerator for at least two hours (and up to three days).
Place the scoops of dough onto the cookie sheet (flat-side down/rounded-side up), spacing the cookies about 2 inches apart. I was able to fit 13 cookies on my pan, alternating rows of 3 cookies and 2 cookies.
Underbaked cookies may sink. Make sure the cookies are puffy and dry looking in the center before pulling them out of the oven.
Be sure to make your glaze thick enough so that it won't run off of the cookies. Start with 1 tablespoon eggnog and add a teaspoon more at a time until the glaze is smooth and just loose enough to drizzle. You can always add more eggnog for a thinner glaze or add back more powdered sugar for a thicker glaze.
The amount of glaze in this recipe is enough to drizzle an entire batch of cookies. You may need to make extra glaze if you wish to "frost" all of the cookies and decorate with sprinkles.
If decorating with sprinkles, sprinkle them immediately after glazing each cookie, because if the glaze starts to set too much, the sprinkles won't stick.
Once the glaze is completely hardened, cookies may be gently stacked for storage. Put a sheet of wax paper between layers to prevent them from sticking together.