In a medium bowl, whisk together the flour, baking powder, salt, and spices. Set aside.
In the bowl of a standing mixer fitted with the paddle attachment, cream together the butter and both sugars until light and fluffy, about 3 minutes.
Add the eggnog, egg yolks, vanilla, and rum extract to the standing mixer bowl. Mix at medium speed until smooth.
With the mixer on low speed, gradually add the flour mixture, mixing just until combined. The dough will be sticky. Cover the dough with plastic wrap and refrigerate for at least one hour and up to 24 hours.
Preheat the oven to 300 degrees Fahrenheit. Line a large baking sheet with parchment paper or silpat.
Using the refrigerated dough, scoop the dough out in heaping tablespoonfuls (or use a medium cookie scoop), placing them 2 inches apart. Bake in preheated oven for 20 minutes or until the edges begin to turn golden.
Let them cool on the pan for 1 to 2 minutes, then transfer to a wire baking rack to cool completely.
Make the frosting: In a medium bowl, combine the powdered sugar, softened butter, rum extract, and 3 tablespoons of eggnog. Mix until smooth, adding one or two additional tablespoons of eggnog to achieve the desired consistency.
Frost the cooled cookies and sprinkle with a pinch of nutmeg and coarse sugar, like turbinado.