Then in a large mixing bowl, whisk together the flour, baking powder, salt, nutmeg, and cinnamon. Set aside.
In the bowl of an electric mixer with a paddle attachment, whip together the butter, sugar and brown sugar until light and fluffy. This will take 3-5 minutes. Next, mix in the egg yolks, one at a time. Then add the vanilla and rum extract and mix until blended.
With your mixer on low, slowly add about half of the dry ingredients, then the eggnog, then the rest of the dry ingredients. Mix until just combined.
Place heaping tablespoons of the dough about two inches apart on a baking sheet lined with parchment paper. Bake for 11-14 minutes. Allow the cookies to rest on the baking sheet several minutes before moving them to a wire rack to cool.
To make the frosting, in the bowl of an electric mixer with a whisk attachment, whip the butter until light and fluffy. Add the rum extract, vanilla extract, salt and about three tablespoons of eggnog. Then, gradually mix in the powdered sugar. Add additional eggnog to reach the desired consistency.
After the cookies have cooled completely, frost them with Eggnog Frosting. Sprinkle lightly with cinnamon before the frosting sets.
Cookies can be stored in an airtight container. If stacking, I recommend refrigerating the cookies first to harden the frosting. Then stack using wax paper in-between each layer.