Preheat oven to 350°F (175°C). Grease a standard loaf pan and line with a parchment sling if desired.

Brown the butter: Melt unsalted butter over medium heat in a light-colored saucepan until golden with toasty brown flecks and a nutty aroma, 5–7 minutes. Transfer to a heatproof bowl to cool slightly.

Mash bananas in a large bowl until mostly smooth with a few small chunks for texture.

Whisk in the slightly cooled browned butter, granulated sugar, eggs, and vanilla until glossy and well combined.

In a separate bowl, whisk together flour, baking soda, and salt to evenly distribute leavening.

Gently fold the dry ingredients into the wet mixture just until no dry spots remain. Do not overmix.

Scrape batter into the prepared pan. Smooth the top; optionally sprinkle coarse sugar or add a sliced banana on top.

Bake on the center rack for about 60 minutes, or until a toothpick inserted in the center comes out clean or with a few moist crumbs. Tent with foil if browning too quickly.

Cool in the pan 10–15 minutes, then lift out to a rack to cool completely before slicing.