Preheat oven to 350°F. Line two baking sheets with parchment.
Whisk flour, cream of tartar, baking soda, salt, nutmeg, and cinnamon in a bowl.
Cream softened butter, granulated sugar, and brown sugar until light and fluffy, about 2 minutes.
Beat in eggnog, vanilla, and rum extract until smooth. Scrape the bowl.
Mix dry ingredients into the wet on low just until a soft dough forms. Do not overmix.
Stir together the rolling sugar, cinnamon, and nutmeg in a shallow dish.
Scoop 1½-tablespoon portions, roll into balls, and coat in the spiced sugar.
Arrange 2 inches apart on sheets and gently flatten to about ¾-inch thick.
Bake 9–11 minutes, until edges are set and centers look soft.
Cool 5 minutes on the sheet, then transfer to a rack to finish cooling.