• Preheat the oven to 425° degrees F. Lightly grease a non-stick 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 3 tablespoons of eggnog and 1 tablespoon of rum into the powdered sugar.
• If the glaze is too thick, beat in more eggnog, a few drops at a time. If frosting becomes too thin, beat in a small amount of powdered sugar.
• Dip each doughnut halfway into the glaze and allow the glaze to set.