Step 1: Preheat your oven to 350°F (175°C). Lightly grease a 12-cavity donut pan with butter or baking spray. Set aside.
Step 2: In a large bowl, whisk together the flour, granulated sugar, baking powder, baking soda, ground nutmeg, and salt. Make a well in the center.
Step 3: In a separate medium bowl, whisk together the egg, milk, melted butter, and vanilla extract until well combined.
Step 4: Pour the wet ingredients into the well of the dry ingredients. Stir with a spatula until just combined. Be careful not to overmix; a few lumps are fine.
Step 5: Transfer the batter to a large piping bag or a Ziploc bag with a corner snipped off (about 1/2 inch opening). Pipe the batter evenly into the prepared donut cavities, filling each about two-thirds full.
Step 6: Bake for 12-15 minutes, or until the donuts are golden brown and spring back when lightly touched. A toothpick inserted into a donut should come out clean.
Step 7: Remove the pan from the oven and let the donuts cool in the pan for 5 minutes before transferring them to a wire rack to cool completely.
Step 8: While donuts cool, prepare the glaze: In a medium bowl, whisk together the powdered sugar, 3 tablespoons of milk, and vanilla extract until smooth. Add more milk, 1 teaspoon at a time, if a thinner consistency is desired.
Step 9: Once the donuts are slightly warm but not hot, dip each donut into the glaze, letting any excess drip off. Place them back on the wire rack for the glaze to set. Serve immediately or store as directed.