Line 12 cup muffin pan with paper liners.Peel bananas and place in large bowl with maple syrup and rum extract. Mash until bananas are liquidy but still slightly chunky.
Place the butter in a heavy saucepan and cook, stirring occasionally, until golden brown. Pay close attention to the butter or it will easily burn on you.
Add about 1 cup of the 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.
Add butter/banana mixture to the remaining banana mixture in bowl. Whisk in egg. Fold in flour until just combined. Do not overmix.
With a large ice cream scoop, evenly divide batter between 12 muffin cups. Bake for approximately 25 minutes or until toothpick comes out clean when inserted into center of muffin.