Prep: Preheat the oven to 350°F. Line a baking sheet with parchment paper and set aside.
Flour mixture: In a large bowl, stir together flour, baking soda, cinnamon, nutmeg, and salt. Set aside.
Butter mixture: In a large mixing bowl, use an electric hand mixer to beat the softened butter until very creamy. Add the granulated sugar and brown sugar and beat until creamy and fluffy, about 3 minutes. Add the egg and beat until combined. Finally, add the eggnog and vanilla extract and beat until combined.
Combine flour and butter mixtures: Add the flour mixture to the butter mixture and mix on low speed until just combined. Switch to a spatula to finish mixing the dough at the end when it becomes too thick.
Scoop dough: Use a small/medium cookie scoop to scoop the cookie dough. Release the cookie dough balls onto your lined baking sheet, leaving about 2" in between each cookie.
Bake: Bake cookies for about 8-10 minutes or until the bottom edges just begins to lightly brown. The cookies will look puffy but will deflate and flatten as they cool.
Cool: Let the cookies cool on the baking sheet for about 5 minutes before transferring them to a wire rack to cool completely.
Eggnog Buttercream
Beat mixture: In a large mixing bowl, use an electric hand mixer to beat softened butter until it becomes very creamy and pale in colour. Add the powdered sugar and nutmeg and beat until smooth and creamy, about 5 minutes, scraping down the bowl at least once.
Add liquids: Add the eggnog and rum and beat until creamy and fluffy, about another 5 minutes, scraping down the bowl as needed.
Pipe buttercream: Transfer the buttercream to a piping bag and cut the tip of the bag off. Pipe a generous dollop of buttercream in the middle of one cookie. Gently press a matching cookie on top until the buttercream squishes out to but not past the edges of the cookies.