Preheat your oven to 350°F (175°C) and grease a donut pan.
In a large bowl, whisk together the flour, sugar, baking powder, baking soda, salt, and nutmeg.
In another bowl, mix the buttermilk, eggs, melted butter, and vanilla extract until well combined.
Pour the wet ingredients into the dry ingredients and stir until just combined. Do not overmix.
Spoon the batter into the prepared donut pan, filling each cavity about 2/3 full.
Bake for 10-12 minutes or until the donuts spring back when lightly touched.
Allow the donuts to cool in the pan for 5 minutes before transferring them to a wire rack to cool completely.
For the glaze, whisk together the powdered sugar, milk, and vanilla extract until smooth.
Dip the cooled donuts into the glaze and let them sit on the wire rack for a few minutes to set.