In a small saucepan, heat the butter over low heat for roughly 5 minutes, swirling the pan every 20 seconds or so. Heat until the milk solids turn a light golden brown color. Remove from heat and allow to cool slightly.
In a large bowl, mix together sugar, mashed bananas, and brown butter. After mixing, stir in eggs, yogurt, and vanilla.
In a separate bowl, whisk both flours, baking soda, and salt. Fold gently into batter until just combined.
Pour into loaf pan and top with thinly sliced banana pieces. I used ½ a banana for this part. Bake for roughly 1 hour, until a knife inserted in the center of the loaf comes out clean. Cool on a wire rack before serving!