Whisk flour, baking powder, salt, nutmeg, and cinnamon together in a large bowl.
Whisk eggnog, eggs, oil, and rum extract together in a small bowl. Add to flour mixture and mix briefly to combine. Batter should be slightly lumpy.
Grease a griddle lightly with cooking spray and heat over medium-high heat. Pour 1/4 cup batter onto the griddle and cook until bubbles form and the edges are dry, 2 to 3 minutes. Flip and cook until browned on the other side, 1 to 2 minutes. Repeat with remaining batter.