Using a stand mixer (or hand mixer + large bowl), cream together the butter, 1/2 cup granulated sugar, and brown sugar on medium speed until light and fluffy, about 3-5 minutes.
3/4 cup salted butter, 3/4 cup granulated sugar, 1/2 cup light brown sugar
Drop mixer speed to low and mix in the egg, eggnog, vanilla extract, and rum extract until incorporated and smooth, about 2-4 minutes.
1 medium egg, 1/4 cup eggnog, 1 teaspoon vanilla extract, 1/2 teaspoon rum extract
Keeping speed on low, quickly scoop in the dry ingredients, adding about 1/3 to 1/2 cup at a time. Once added, continue to mix until dry ingredients first appear to be incorporated; be careful not to overmix.
Add the remaining 1/4 cup granulated sugar to a small bowl. Using a cookie scoop, scoop out about 1-2 tablespoons of cookie dough and roll into a ball (about 1 inch wide) then toss the cookie ball in the bowl of sugar, covering the outside. Place finished cookie ball on prepared baking sheet. Repeat this step until all dough is used, placing cookies about 2 inches apart on the baking sheet.
Bake cookies for 9-11 minutes or until the top of the cookie begins to crinkle and the bottom sides turn a light golden brown.
Let cookies rest on the baking sheet for 5-10 minutes, then transfer to a wire cooling rack to cool completely.
For the Eggnog Icing
While cookies cool, add powdered sugar, eggnog, nutmeg, and cinnamon to a medium bowl. Whisk until consistency is smooth with no clumps.