Preheat the oven to 375°F / 190°C and lightly grease a donut pan.
Whisk together flour, granulated sugar, baking powder, and nutmeg in a large bowl.
In a separate bowl, whisk eggnog, egg, and melted butter until smooth, then fold into the dry ingredients until just combined.
Spoon the batter into the donut pan, filling each cavity about two-thirds full, and bake for 10 to 12 minutes until golden and a toothpick comes out clean.
Let the donuts cool in the pan for 3 minutes, then transfer to a wire rack.
Whisk powdered sugar and milk until smooth to make the glaze.
Dip the top of each cooled donut into the glaze and let set for 5 minutes before serving.