Heavily grease the donut pan with butter or cooking spray. Set aside.
Preheat the oven to 350°F/180°C.
In a large bowl, mix the flour, sugar, baking powder, and salt.
In a medium bowl, whisk the egg and then add and whisk milk, vanilla, and melted butter.
Add the wet ingredients to the dry ingredients and stir until just combined. The batter is going to be very thick!
Using a piping or Ziploc bag, evenly divide the batter into the six donut holes. Cut a corner off the bottom of the bag and pipe the batter into each donut cup, filling 2/3 to 3/4 of the way full.
Bake for 10-14 minutes or until the edges are lightly browned. Transfer to a baking rack and allow to cool before glazing.