For the Cookies: Heat the oven to 350°F. Position rack to lower third of the oven. Prepare cookie sheets by either lining them with parchment paper, a Silpat or greasing them with butter.
Sift the flour, baking powder, and salt together into a bowl. Stir in the nutmeg and cinnamon.
Beat the butter in a mixing bowl, using either a hand mixer or a standing mixer fitted with the paddle attachment, until light and creamy. Add the sugars and mix until smooth; add the egg yolks blending until combined. Add the vanilla extract, rum extract, and eggnog. Mix well. Turn the mixer to low and slowly add the dry ingredients, mix until combined.
Drop dough by heaping teaspoon or using a small (1-inch) ice cream scooper on to the prepared cookie sheets spacing them 2 inches apart. Bake in the oven 12-13 minutes, or until cooked and lightly browned. Let cool on cookie sheet for 5 minutes, then remove to wire racks to cool completely.
For the Frosting: Whip the softened butter with a hand mixer or a standing mixer fitted with the paddle attachment until very pale. Add the rum extract and 3 tablespoons of eggnog; mix well. Mix in the powdered sugar and mix until smooth. Add more eggnog if the frosting seems too stiff. It should be fairly soft.
Once the cookies are completely cooled, frost with the frosting. Allow to sit on wire racks until the frosting has set and is firm. Store cookies in an airtight container. Place parchment paper between layers if stacking.