Mix the flour, sugar, baking powder, baking soda, salt, and nutmeg if using, then stir in the buttermilk, eggs, melted butter, and vanilla until a soft dough forms.
Roll the dough to about ½ inch thick on a floured surface and cut into donut shapes, rerolling scraps as needed.
Fry donuts at 350°F or 175°C for 1 to 2 minutes per side until golden, or bake at 425°F or 220°C for 8 to 10 minutes in a greased donut pan.
Whisk powdered sugar, milk, and vanilla for the glaze, then dip or drizzle over warm donuts and let set before serving.