Preheat your oven to 350°F (175°C). Grease a donut pan.
In a mixing bowl, whisk together the flour, baking powder, baking soda, and salt.
In another bowl, mix the sugar, mashed bananas, milk, egg, and vanilla extract. Add the melted butter and mix well.
Combine the wet and dry ingredients, mixing until just combined.
Spoon the batter into the donut pan, filling each cavity about ¾ full.
Bake for 15 minutes, or until a toothpick inserted into the donuts comes out clean. Let them cool in the pan for a few minutes before transferring to a wire rack to cool completely.
For the glaze, melt the chocolate chips and butter together in a microwave-safe bowl, stirring every 30 seconds until smooth. Stir in the milk and vanilla extract.
Dip the cooled donuts into the chocolate glaze. Let the glaze set before serving.