Preheat your oven to 350 degrees F (175 degrees C) and line a baking sheet with parchment paper, silpat mat, or grease using non-stick cooking spray.
In a large bowl, cream together the unsalted butter and sugar until smooth. Add the egg and combine until smooth again, then add the eggnog and mix until smooth.
Add vanilla extract, baking soda, salt, cream of tartar, and ground ginger then mix until evenly distributed through the dough mixture. Add flour and mix until the flour is worked into the wet mixture to form your cookie dough.
Scoop the dough in heaping tablespoonfuls and roll in the palms of your hand to make a 1 ½" ball then place onto your lined or greased baking sheet with about 2" of spacing between the cookies and pan edges. Press down gently on each dough ball to flatten slightly before baking.
Bake at 350 degrees F (175 degrees C) for 8-10 minutes or until the bottom edges start to show a touch of golden color. Try a test batch of a few cookies before baking the rest so that you can find the best baking time for your oven.
Remove from oven and allow cookies to cool for 2-3 minutes on the baking sheet, then transfer to a wire cooling rack and allow to cool completely.
Nutmeg Frosting Instructions
In a medium mixing bowl or your stand mixer, whip the softened butter until fluffy. Add the nutmeg and gradually add the eggnog until it is all added (which can get messy, so start handmixers or stand mixers on low). Add the sifted powdered sugar and mix until the frosting is light and creamy, about 2-3 minutes.
Spoon the frosting onto the center top of each of the cooled cookies and spread outward toward the edges in circular motion. Top with optional garnish (coarse sugar and freshly grated nutmeg).