In a small saucepan over medium heat, melt the butter and cook, stirring frequently, until it turns golden brown and smells nutty. Immediately pour into a bowl to stop the cooking. Cool slightly.
Preheat oven to 350°F (175°C). Line a loaf pan with parchment paper or lightly grease with baking spray.
In a large bowl, mash the bananas until smooth. Whisk in the browned butter, granulated sugar, brown sugar, sour cream, eggs, and vanilla until fully combined.
In a separate bowl, whisk together the flour, cinnamon, baking powder, baking soda, and salt. Gently fold the dry mixture into the wet ingredients until just combined.
Pour batter into prepared pan and smooth the top. Use a knife to make a slice into the top of the batter about 1/2” to get that split appearance in the top. Then bake for 55–65 minutes, or until a toothpick inserted in the center comes out mostly clean (a few moist crumbs are fine).
Let cool in the pan for 10 minutes, then transfer to a rack and cool completely before slicing (at least 1 hour).