Preheat the oven to 350°F / 175°C and grease a donut pan with cooking spray.
Whisk together flour, sugar, baking powder, and nutmeg in a large bowl.
In a separate bowl, whisk together eggnog, egg, and melted butter, 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 12 to 14 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 together powdered sugar and eggnog until smooth to make the glaze.
Dip the top of each donut into the glaze and let set for 5 minutes before serving.