Heat oven to 350°F. Grease a 13×18-inch rimmed sheet pan and line with parchment.
Whisk flour, sugar, baking powder, baking soda, salt, nutmeg, and cinnamon in a large bowl.
Warm melted butter and eggnog in a saucepan over medium heat until steaming; do not boil. Remove from heat.
Pour the warm mixture into the dry ingredients and whisk until mostly smooth.
Whisk in sour cream, vanilla, and rum extract to slightly cool the batter.
Beat in eggs one at a time just until incorporated; do not overmix.
Spread batter evenly in the pan. Tap pan once or twice to release bubbles.
Bake 18–22 minutes, until the top springs back and a toothpick comes out clean or with a few moist crumbs.