Preheat oven to 350° F. Lightly grease a donut tin with nonstick cooking spray. Set aside.
In a medium pan over low heat, melt the butter for about 7 minutes, stirring occasionally. The butter will turn from yellow to golden to light brown. A nutty aroma will arise, which is the cue to remove the butter from the heat and transfer to a heat-safe bowl. Cool slightly.
In a medium bowl, whisk together flour, cinnamon, baking powder and salt. Set aside.
In a large mixing bowl, vigorously whisk granulated sugar, brown sugar, egg, vanilla, brown butter and mashed bananas (you can also use a hand mixer). Add the flour mixture and use a rubber spatula to stir the batter just until combined. Scrape down the sides and bottom of the bowl to find any flour pockets. Avoid over mixing.
Transfer the batter to a piping bag and pipe batter into prepared donut tin. Alternatively, you can spoon the batter into the pan. Fill each donut cavity about three-quarters full.
Bake at 350° F for 12 to 15 minutes, or until a toothpick inserted in a donut comes out clean. Cool donuts in the tin for 5 minutes, then transfer to a wire cooling rack. Prepare glaze.
Brown Butter Glaze
Brown another 1/4 cup of butter, following the same steps for the donuts. Set aside to cool slightly.
Whisk together brown butter, powdered sugar and 1 tablespoon milk. Once the ingredients are combined, add more milk only if you would like a thinner consistency (I used about 1 1/2 tablespoons).
Dip the top of a warm donut in the glaze and set on a plate. Dust with cinnamon or nutmeg if desired and serve!
Notes
Browning butter: This is a simple step that adds a subtle nutty flavor. If you prefer, you can use melted butter instead although the nutty flavor won't be present.Bananas: Use ripe bananas that are brown on the peel. They are sweeter and will add more flavor.Muffin tin: If you don't have a donut tin, use a standard 12-cup muffin tin. Fill each cup 3/4 full and bake at 375° F for 15 to 18 minutes.Store in an airtight container for up to 3 days. Donuts are best when eaten the day they are baked.Freeze unfrosted donuts in an airtight container for up to 2 months. Defrost in the refrigerator before bringing to room temperature and adding glaze.