Preheat the oven to 350 degrees and line baking sheets with parchment paper.
In a large mixing bowl, beat the softened butter, granulated sugar, and brown sugar together until the mixture is light, fluffy, and well combined.
Beat in the egg and vanilla extract.
In a separate large bowl, thoroughly mix together the flour, cream of tartar, baking soda, and kosher salt.
Gradually add the flour mixture and eggnog to the creamed butter mixture, alternating between the two. Mix until well combined.
In a large plate, mix together ⅓ cup sugar, ground nutmeg, and cinnamon for rolling the cookies.
Scoop rounded tablespoons of dough and roll them in the sugar and spice mixture until coated.
Place the coated dough balls 2 inches apart on the prepared baking sheets.
Bake for 11 minutes, or until the cookies are set and begin to show slight cracks on the surface.
Remove the cookies from the oven and let them cool completely on a wire rack.
Optional glaze: In a small bowl, mix together the powdered sugar, milk, and vanilla extract until smooth. If the glaze is too thick, add another tablespoon of milk.
Pour the glaze into a small zip-top bag and cut a small tip off one corner. Squeeze the glaze onto the cooled cookies in a decorative pattern.
Allow the glaze to harden for about 20 minutes.
Enjoy these delightful eggnog snickerdoodle cookies!