Whisk together the flour, baking powder, baking soda, salt, nutmeg, and cinnamon in a medium bowl until well combined.
In a large bowl, beat the softened butter and sugar together until light and fluffy, about 3-4 minutes.
Beat in the eggs one at a time, then add the eggnog and vanilla extract. The mixture might look slightly curdled – that's normal.
Gradually add the dry ingredients to the wet ingredients, mixing on low speed just until combined. Do not overmix.
Cover the dough and refrigerate for at least 30 minutes.
Preheat your oven to 350°F (175°C) and line baking sheets with parchment paper.
Scoop tablespoon-sized balls of dough and place them about 2 inches apart on the prepared baking sheets.
Bake for 10-12 minutes until the edges are just set and the centers are still soft. The cookies should have a pale golden hue.
Allow cookies to cool completely on the baking sheets.
For the frosting, beat the softened butter until creamy. Gradually add the powdered sugar, then mix in the eggnog, vanilla, nutmeg, and salt. Adjust consistency as needed.
Once the cookies are completely cool, spread a generous layer of eggnog frosting on each one. For a festive touch, sprinkle with additional nutmeg or colored sugar.
Let the frosting set for about 20 minutes before stacking or storing.