Heat the oven to 350°F and grease a standard 12-cavity doughnut pan.
Whisk the flour, sugar, baking powder, cinnamon, and salt in a bowl.
Whisk the milk, egg, melted butter, and vanilla in a second bowl. Pour the wet mixture into the dry ingredients and stir just until no dry streaks remain.
Pipe or spoon the batter into the pan, filling each cavity about two-thirds full.
Bake for 10 to 12 minutes, until the doughnuts spring back lightly when touched and a toothpick comes out clean. Cool in the pan for 5 minutes, then transfer to a rack.
Whisk the powdered sugar, milk, and vanilla until smooth. Dip the cooled doughnuts into the glaze and let the excess drip back into the bowl.