Preheat your oven to 350°F (175°C). Grease a 9x5 inch loaf pan.
In a saucepan, melt the butter over medium heat. Continue to cook, swirling the pan occasionally, until the butter turns golden brown and smells nutty. Remove from heat and let it cool slightly.
In a large mixing bowl, whisk together the browned butter and sugar until well combined.
Add the eggs, one at a time, beating well after each addition. Stir in the mashed bananas, milk, and vanilla extract.
In a separate bowl, whisk together the flour, baking soda, and salt. Gradually add the dry ingredients to the wet ingredients, mixing just until combined.
Pour the batter into the prepared loaf pan and smooth the top.
Bake for 60 minutes, or until a toothpick inserted into the center comes out clean. Let the bread cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.