Preheat oven to 350°F (175°C) and lightly grease a donut pan.

In a medium bowl, whisk together flour, sugar, baking powder, baking soda, and salt.

In another bowl, whisk together buttermilk, melted butter, egg, and vanilla extract until smooth.

Add the wet ingredients to the dry ingredients and stir gently until just combined.

Spoon or pipe 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 pressed.

Allow donuts to cool in the pan for 5 minutes before transferring to a wire rack.

For the glaze, whisk together powdered sugar, milk, and vanilla extract until smooth.

Dip cooled donuts into the glaze, letting excess drip off.

Place on a wire rack until the glaze sets.