Brown the butter in a saucepan over medium heat until golden and nutty-smelling, about 4–5 minutes. Cool for 10 minutes.
Preheat the oven to 350°F (175°C). Grease a 9×5-inch loaf pan.
Mash the bananas in a large bowl until mostly smooth.
Whisk in the browned butter, brown sugar, and granulated sugar. Add the eggs one at a time, then stir in the yogurt and vanilla.
Fold in the flour, baking soda, salt, and cinnamon just until combined.
Fold in nuts or chocolate chips, if using.
Pour into the prepared pan and bake for 55–60 minutes, until a toothpick comes out clean.
Cool in the pan for 15 minutes, then transfer to a wire rack to cool completely before slicing.
