Preheat oven to 375 degrees. Prepare metal doughnut pans by greasing with butter and dusting with flour. Set aside.
In a large bowl sift flour, sugar, baking powder, cinnamon, nutmeg and salt. In a second bowl, break eggs then whisk in egg nog, and melted butter. Fold the wet ingredients into the dry ones until just combined to avoid ‘tough’ doughnuts.
Spoon batter ¾ full into doughnut moulds. Bake about 10 minutes or until a toothpick comes out clean. Set pans on a wire rack to cool for a few minutes, then transfer doughnuts to a wire rack to cool completely.
Make the glaze by whisking powdered sugar, nutmeg, cinnamon, vanilla and egg nog to a smooth, slightly runny consistency. Thin with a little milk if necessary.
One by one, invert each doughnut, dipping top surface into glaze, wiggling a bit, then turning 90 degrees to allow excess glaze to run off. Return to wire rack to set.
Garnish with sprinkles to add a festive look. Enjoy!