In a mixing bowl, stir together the flour, baking powder, baking soda, salt, and nutmeg. Set aside.
In the bowl of a stand-mixer fitted with the paddle attachment, cream together the butter and sugar until light and fluffy. Incorporate the eggs, followed by the rum extract. In a small bowl, stir together the Greek yogurt and the eggnog until combined, then add it to the bowl of the stand mixer, scraping the sides of the bowl as needed as you mix. Gradually add the dry ingredients until it all just comes together. The dough will be sticky.
Divide the dough in half and flatten each half into a disc. Wrap each disc in plastic wrap and refrigerate at least two hours.
Preheat oven to 425, placing oven rack in the middle of the oven. Line cookie sheets with parchment paper.
Unwrap the dough and place it onto a floured work surface. Dust the top of the dough with flour also. Roll the dough to 1/4-inch thickness. Cut the dough with a round cutter (about 2 inch size). Place the cookies on the prepared baking sheet and repeat with the remaining dough.
Bake the cookies 6-7 minutes until cooked through and golden on the bottom (mine were perfect at 6 minutes. Be careful not to over bake. You don't want any color on the top of sides of the cookies, just the bottom). Transfer cookies immediately to a cooling rack to cool completely before frosting.
To Make Frosting:
In the bowl of a stand-mixer fitted with the paddle attachment, cream together the butter and rum extract. Reduce speed and add the powdered sugar gradually, scraping the sides of the bowl as needed. Add the eggnog gradually, then increase the speed and beat frosting until light and fluffy, about 1-2 minutes. Frost cookies (I used a piping bag with a large round tip in a round swirling motion, then used a small off-set spatula to lightly spread the frosting). Sprinkle the cookies with turbinado sugar and/or freshly ground nutmeg directly after frosting.
Store cookies in an airtight container.