Preheat the oven to 350 F (177 C). Grease a 6 ct. donut pan or muffin pan (see expert tips).
In a large bowl, whisk the flour, baking powder, ground cinnamon and salt together until well combined.
In another large bowl, pour the light brown sugar, eggs, mashed bananas, melted butter, and vanilla extract. Whisk well to fully combine.
Add the dry ingredients to the wet. Whisk well until smooth and fully combined with no lumps.
Spoon or pipe about three heaping tablespoons of banana donut batter into each of the prepared donut cavities in the donut pan. This should fill about ¾ of each cavity of the donut pan.
Bake the donuts for about 10-12 minutes. The donuts are done when the tops bounce back quickly when lightly pressed. Cool the donuts in the pan for 10 minutes before gently taking them out to cool on a cooling rack. While the banana doughnuts cool, make the brown butter glaze.
Make the brown butter by melting the butter in a medium sized pot on low heat. Allow the melted butter to begin to foam. Once it starts to foam whisk or stir the liquid butter frequently. The butter will begin to smell nutty and very fragrant. Once the foam has subsided, the butter should have some brown, lightly toasted bits floating at the bottom. Pour the brown butter into a bowl, leaving as much of the brown bits in the pan as possible. Whisk in the powdered sugar, vanilla extract and teaspoon of room temperature milk. Whisk well until smooth and combined. Add more milk as needed until the brown butter icing is to your liking.
Dip the cooled banana cake donut into the brown butter glaze for donuts. Gently shake off any excess icing. Place each iced banana donut on the cooling rack. After setting for about 1-2 minutes, top with banana chips if desired. Allow the donuts to set for about an hour before eating. Enjoy!