In the bowl of a stand mixer fitted with the paddle attachment, beat the butter and sugars until light and fluffy, about 3-4 minutes. You may need to pause the mixer and scrape down the sides of the bowl. Add the rum and vanilla and mix for 30 seconds.
Add the eggs and yolks and beat until fluffy, about 2 minutes. In a separate bowl, whisk together the flour, cream of tartar, baking powder, salt, cinnamon, and nutmeg. Add the dry ingredients to the mixer and mix on low. Add the eggnog while the dry ingredients are mixing.
Mix until the dough starts to clump together. Using a 2 oz scoop, portion the dough onto a baking sheet lined with parchment paper. Cover with plastic wrap and chill for 1 hour.
Preheat an oven to 350 F. In a small bowl, whisk together the sugar, cinnamon, and nutmeg. Roll the cookie balls in the sugar mixture until well coated and place them on a baking sheet lined with parchment paper. Give the cookies at least 2-3 inches of space as they will spread a bit (you may need two trays).
Bake one tray at a time (keeping the other in the fridge) for 15-17 minutes, until the cookies are just barely turning golden brown and are puffy in the middle. Cool on a wire rack. Repeat with the second tray. Cool slightly before serving.