Combine butter with sugar and beat on medium speed until well creamed together.
Add eggs and vanilla extract. Mix into butter and sugar until eggs are well beaten.
In a separate bowl, combine flour, baking soda, cream of tartar and nutmeg. Slowly add into batter and beat on medium low speed until well combined and dough forms.
Refrigerate dough for 30 minutes.
Use a two-table spoon cookie scoop and space evenly on a cookie sheet lined with a Silpat or parchment paper. Bake at 350°F for 10-12 minutes in the middle rack of your oven.
Allow butter to come to room temperature. Using a stand mixer, beat the butter for several minutes until light and fluffy.
Add powdered sugar one cup at a time and slowly increase speed from low to medium. As the frosting becomes thicker, slowly add eggnog and slowly increase speed to medium high; beat for several minutes until light and fluffy.
Spread over cooled cookie and sprinkle with nutmeg.