Preheat your oven to 350°F (175°C). Grease a donut pan lightly or use a silicone mold.
In a large mixing bowl, combine the all-purpose flour, granulated sugar, baking powder, ground nutmeg, ground cinnamon, and salt. Whisk them together thoroughly.
In a separate bowl, whisk together the eggnog, large egg, vanilla extract, and melted unsalted butter until smooth.
Pour the wet ingredients into the bowl with the dry mixture. Stir them together until just combined.
Using a piping bag or a spoon, fill each cavity of the donut pan approximately two-thirds full with the batter.
Place the filled donut pan into the preheated oven. Bake for 10-12 minutes until golden brown.
Remove the donuts from the oven and let them cool in the pan for about 5 minutes, then transfer to a wire rack.
In a mixing bowl, whisk together the powdered sugar and milk until smooth to create the glaze.
Once the donuts are completely cool, dip each one into the glaze and return them to the wire rack.
Lightly dust the glazed donuts with additional ground nutmeg before serving.
Notes
For an inviting display, arrange the donuts on a festive plate or elevated stand.