Preheat the oven to 350°F (175°C) and lightly grease a doughnut pan.
In a medium bowl, whisk together the flour, sugar, baking powder, cinnamon, and salt.
In a separate bowl, whisk the egg, milk, melted butter, and vanilla until combined.
Add the wet ingredients to the dry ingredients and stir just until no dry streaks remain. Do not overmix.
Spoon or pipe the batter into the prepared doughnut pan, filling each cavity about 2/3 full.
Bake for 10 to 12 minutes, or until the doughnuts spring back when lightly touched.
Let the doughnuts cool in the pan for 5 minutes, then transfer to a wire rack.
While still warm, brush each doughnut with melted butter.
Mix the coating sugar and cinnamon in a shallow bowl, then roll each doughnut in the mixture until well coated.
Serve warm and enjoy.