In a large bowl with an electric mixer or in a stand mixer, beat the softened butter, granulated sugar, and brown sugar on medium speed until light and fluffy, about 2-3 minutes.
Add the egg and vanilla extract, mixing until just combined. Scrape down the sides of the bowl.
In a separate medium bowl, whisk together the flour, baking powder, cream of tartar, cinnamon, nutmeg, and salt.
With the mixer on low, gradually add the dry ingredients to the butter mixture in two parts, mixing until just a few streaks of flour remain.
Slowly stream in the eggnog and mix on low until the dough is just combined. Do not overmix. Cover the bowl and chill the dough in the refrigerator for at least 1 hour, or up to 2 days.
Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
In a small, shallow bowl, stir together the 1/3 cup granulated sugar, 3/4 teaspoon cinnamon, and 1/4 teaspoon nutmeg for the coating.
Scoop the chilled dough into 1-inch balls. Roll each ball between your hands until smooth, then roll it in the cinnamon-sugar mixture to coat completely.
Place the dough balls on the prepared baking sheets, about 2 inches apart. Bake for 12-14 minutes, or until the edges are set and the centers look soft. They will puff up and then settle as they cool.
Let the cookies cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.