In a medium saucepan over medium heat, melt the butter. Once fully melted, it will begin to foam. Stir constantly with a rubber spatula or wooden spoon so you can see the color change. When the butter turns a deep golden brown and smells nutty, remove it from the heat immediately.

Pour the browned butter into a heatproof bowl and let it cool for 15 to 20 minutes. You should have about 6 tablespoons (85 to 90 grams) of browned butter to use.

Preheat the oven to 325°F. Line a 9x5-inch loaf pan with parchment paper, leaving some overhang on the long sides for easy removal.

In a large bowl, combine the mashed bananas, granulated sugar, brown sugar, eggs, vanilla, milk, sour cream, and the cooled browned butter. Whisk until smooth and well blended.

Add the flour, baking soda, and salt to the wet ingredients. Using a spatula, gently fold until just combined and no streaks of flour remain. Do not overmix.

Pour the batter into the prepared loaf pan and spread it evenly. Bake for 60 to 65 minutes, or until the top is a rich caramel color and a wooden skewer inserted into the center comes out clean. (If the skewer has wet batter or wet crumbs, bake for 5 more minutes and check again.)

Let the banana bread cool in the pan for 10 to 15 minutes, then lift it out using the parchment paper and transfer to a wire rack to cool completely before slicing.