Line a 12 cup muffin pan with paper liners. Place bananas in a large bowl and mash well, leaving a few chunks.
Add maple syrup and rum extract and stir to combine.
Place the butter in a heavy saucepan and cook over medium heat, stirring occasionally, until golden brown. Pay close attention to the butter or it will easily burn on you.
Add 1 cup of the mashed banana mixture to the browned butter. Simmer a another minute or so until a darker color is achieved. Be very careful when adding the banana mixture to the butter as it will bubble and spatter.
Return butter/banana mixture to the remaining banana mixture in bowl. Whisk in egg until well incorporated. Fold in flour until just combined.
With a large cookie scoop, evenly divide batter into prepared muffin cups.
Bake for 22-25 minutes or until toothpick comes out clean when inserted into center of muffin. Cool on a wire rack.