In the bowl of a stand mixer fitted with the paddle attachment, cream butter and brown sugar together until light and fluffy. Beat in egg yolks, one at a time, followed by eggnog and rum extract. Add flour and salt and beat until just combined. Refrigerate at least 2 hours. 

Preheat oven to 325F. Line 2 baking sheets with parchment paper. 

Take tablespoons of spoon and shape into balls. Place 2 inches apart on prepared baking sheet. 

Bake until bottoms are golden brown, 12 - 15 minutes, rotating pans halfway through. Transfer to a wire rack to cool. Repeat with remaining dough. Cool completely before frosting. 

To make the frosting, beat butter, powdered sugar, rum extract, cinnamon, and nutmeg until combined. Add enough eggnog to reach desired consistency. Spread on top of cooled cookies and sprinkle with extra nutmeg.