Preheat the oven to 350°F (175°C) and grease a donut pan.
Whisk flour, brown sugar, baking powder, cinnamon, and salt.
In another bowl, combine the egg, milk, butter, and vanilla.
Stir the wet ingredients into the dry mixture just until combined.
Pipe or spoon the batter into the donut cavities.
Bake for 10–12 minutes, then cool completely.
Stir powdered sugar, maple syrup, and enough milk to make a smooth glaze.
Dip each donut into the maple glaze and allow it to set.