In a large mixing bowl, whisk together the flour, cream of tartar, baking soda, cinnamon, nutmeg, and salt. Set aside.
In the bowl of a stand mixer fitted with a paddle attachment, beat the butter until smooth, about 1 to 2 minutes. Add both sugars and continue beating on medium speed until smooth and creamy.
Scrape down the sides of the bowl, add the egg, and beat just until combined. Add the eggnog and vanilla and beat on medium-high speed until light and fluffy.
Add the flour mixture and beat on low just until combined, about 1 minute.
Cover dough with plastic wrap and refrigerate for at least 2 hours, or up to 5 days. Do not skip this step.
Preheat oven to 375ºF and line a baking sheet with parchment paper or spray with cooking spray.
In a small bowl, combine the sugar and cinnamon.
Using your hands or a small cookie scoop, form dough into 1-inch balls, roll in the cinnamon sugar, then place on the lined baking sheet, at least 2 inches apart. Bake for 8-9 minutes or until edges have set and the tops appear slightly undercooked. Cookies will firm up as they cool. Allow cookies to cool on the baking sheet for at least 5 minutes before transferring to a cooling rack to cool completely.
Cookies will stay fresh in an airtight container at room temperature for up to 1 week.