Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper.
In a medium bowl, whisk together flour, salt, and nutmeg. Set aside.
In a large bowl, cream together the butter, granulated sugar, and brown sugar until fluffy.
Mix in the egg yolk, vanilla, rum extract, and eggnog until well combined.
Gradually add the dry ingredients to the wet mixture until a dough forms.
Roll dough into 1-inch balls and place on the prepared baking sheet.
Use your thumb or the back of a spoon to make an indentation in each ball.
Bake for 12–14 minutes or until edges are lightly golden. Let cool completely.
For the frosting, whisk together powdered sugar, eggnog, and rum extract until smooth.
Spoon or pipe the frosting into the center of each cooled cookie.
Optional: Sprinkle a pinch of nutmeg on top before serving.