Preheat the oven to 350°F/180°C. Grease or line an 8.5x4.5-inch or 9x5 inch loaf pan with parchment paper.
Let the butter melt in a small saucepan over medium heat. Once melted, let it cook for around 3-4 minutes until the milk solids have caramelized and the butter smells nutty. Pour the browned butter into a large bowl and cool down for 10 minutes.
Mash the two bananas with a fork and add them to the cooled butter.
Add in the brown sugar, eggs, and vanilla. Whisk it well together until all combined.
Mix the flour, salt, and baking soda in a separate bowl.
Tip the dry ingredients into the wet ingredients and the yogurt and fold with a rubber spatula until well combined but not overmixed. You will see the brown flecks of butter throughout the batter.
Spoon the mixture into the prepared loaf pan.
Slice the extra banana lengthwise and place the sliced banana on top of the batter. Sprinkle with a bit of turbinado sugar.
Bake in the preheated oven for approximately 60-70 minutes or until the top of the bread springs back when touched and a skewer inserted into the center of the loaf comes out clean. If baking in a 9x5 inch pan it will need less time than a smaller pan.
Remove the baked banana bread from the tin and let it cool on a cooling rack for 20 minutes before slicing.
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Notes
Sweetness - This brown butter banana bread isn't overly sweet. If you want it sweeter, the sugar can be increased to 200g (1 cup.)Brown Butter Banana bread muffins - You can bake this batter in muffins too! Preheat oven to 425°F (220°C) and grease or line a muffin tin with 12 paper liners. Make the batter, then divide over the muffin cups. Bake the muffins for 5 minutes in the preheated oven at 425°F (220°C), then lower the oven heat to 350°F (180°C) and continue baking for 16-18 minutes until golden brown and a cake tester inserted into the center of a muffin comes out clean. The total time will depend on your oven.Cup sizes - The cup sizes given are US-sized. Note that these are smaller than metric. For best results, use grams.Add-ins - Feel free to add nuts, chocolate chips, or dried fruits like cranberries or raisins to the batter for added texture and flavor.Frozen bananas - Feel free to use frozen bananas. Let them thaw and pour off the excess moisture, then mash and use in the recipe.Gluten-free - You can make this quick bread gluten-free using a gluten-free flour blend designed to substitute all-purpose flour directly. Just check if your baking soda and other added ingredients are gluten-free.