Preheat the oven to 350°F. Line a 9-inch loaf pan with parchment paper then grease the paper with cooking spray.
Add the butter to a medium skillet set over low heat. Cook, stirring frequently, until the butter begins to foam and become golden, 5 to 7 minutes. (It will smell nutty and the milk solids will begin to darken.) Immediately transfer the brown butter to a medium bowl and set it aside to cool. (If you leave the butter in the hot pan it will burn.)
In a separate large bowl, whisk together the flour, sugar, brown sugar, baking soda and salt.
Add three of the ripe bananas to the bowl with the brown butter and mash well. Whisk in the sour cream, eggs and vanilla extract.
Add the wet ingredients to the dry and mix just until combined. Transfer the batter into the prepared loaf pan.
Bake the bread until a toothpick inserted comes out clean, 55 to 60 minutes. Remove the bread from the oven and let it cool completely in the pan before slicing and serving.