Preheat oven to 350°F. Spray donut pan with nonstick cooking spray.
In a medium bowl, beat together the sugar egg, bananas, vanilla, and melted butter until well combined. In a small bowl, combine the flour, baking soda, salt, and cinnamon.
Stir the dry ingredients into the wet ingredients until just combined.
Spoon the batter into the donut pan and bake for 13 minutes.
Let cool for 5 minutes before removing from the pan.
While the donuts are cooling, prepare the glaze.
In a small saucepan, bring the brown sugar, water, and salt to a boil. Reduce heat to a gentle simmer and cook, stirring constantly, for 3 minutes.
Remove from the heat and stir in the vanilla.
Let the glaze cool for a few minutes.
Dunk the donuts in the glaze to coat and remove to a cooling rack. The glaze will be very wet and sticky at first, but as it cools it will thicken up slightly.