Preheat the oven to 425 degrees F. Lightly grease a doughnut pan.
In a medium-size bowl, whisk together the flour, sugar, baking powder, cinnamon, nutmeg and salt. In a separate bowl, whisk together the 1/2 cup eggnog, egg, vegetable oil and vanilla. Whisk the wet ingredients into the dry ingredients.
Spoon the batter into the pan, filling each doughnut only 3/4 of the way.
Bake for 7 minutes. Turn the doughnuts onto a cooling rack and allow them to cool.
To make the glaze, whisk the powdered sugar into the 3 tablespoons of eggnog and rum. Dip each doughnut halfway into the glaze and allow the glaze to set.