Melt butter in a small saucepan over medium heat; bring to a low simmer and cook, stirring frequently, until butter is browned, fragrant, and foamy, with brown bits in it, 5 to 10 minutes. Set aside to cool to room temperature, about 15 minutes. Measure 1/3 cup of browned butter; reserve remaining browned butter for another use.

Preheat oven to 350 degrees F (175 degrees C). Grease a loaf pan.

Mash bananas in a large bowl; stir in brown sugar, white sugar, egg, vanilla extract, and 1/3 cup brown butter.

Whisk flour, baking soda, cinnamon, and nutmeg together in a bowl. Fold flour mixture into banana mixture just until all ingredients are moistened; pour into prepared pan.

Bake in the preheated oven until a toothpick inserted near the center comes out clean, about 45 minutes.