Preheat the oven to 350 degrees F. Butter a 9-by-5-inch loaf pan.
Heat the butter over medium-low heat in a small saucepan until the milk solids are deep golden brown and the butter smells toasted, 15 to 20 minutes.
Whisk together the flour, granulated sugar, baking powder, baking soda, cinnamon and salt in a medium bowl.
Stir the brown butter so that any milk solids that have fallen to the bottom are distributed equally throughout the butter. Reserve 4 tablespoons of the brown butter for the glaze. Whisk together the remaining brown butter, brown sugar, eggs and vanilla in a medium bowl, then fold in the mashed bananas.
Fold the banana mixture into the flour mixture until just combined (it's OK if there are some lumps). Transfer the batter to the prepared loaf pan, and bake until the bread is golden brown and a toothpick inserted in the center comes out clean, 1 hour to 1 hour 10 minutes. Run a knife around the edges, and let cool completely in the pan on a rack.
Once the bread has cooled, heat the reserved brown butter in a microwave-safe bowl in the microwave until melted, then whisk in the confectioners' sugar. Let the glaze sit at room temperature until it stiffens up a little but is still pourable, about 5 minutes. Pour and spread the brown butter over the bread. Let set. The bread can be stored at room temperature in an airtight container for up to 3 days.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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