Preheat an oven to 160FAN or 350 degrees Fahrenheit. Butter and flour a loaf pan.
While the oven heats up, brown the butter by heating the butter in a small sauce pan over medium heat until it begins to foam and bubble. Stirring frequently, watch until the butter turns deep amber brown and becomes fragrant - it will smell nutty and rich; this takes between 5 and 10 minutes, depending on your stove top. Once the butter is browned (but not burnt!) turn off the heat.
Combine the mashed bananas, browned butter and vanilla in a bowl.
Separately, combine the flour, brown and white sugar, baking soda, salt and cinnamon. Mix the wet ingredients into the dry ingredients. After the mixtures have started to come together, add the beaten egg and stir until everything is just combined.
At this point, if you want to add nuts or chocolate chips (or anything else that you like in banana bread!) it can be added. I recommend adding about 1/2-3/4 cup of additional ingredients.
Pop the batter into a loaf pan, and decoratively place some banana halves on top, if you wish.
Bake for 50-55 minutes, or until a toothpick or knife inserted into the center of the loaf comes out almost completely clean.
Make sure to check on the load after 20 minutes or so - if the top of the loaf is browned sufficiently, cover it with tin foil while the inside of the bread continues baking.
While the banana bread cools, prepare the glaze by heating the butter and maple syrup in a small sauce pan until the butter has melted completely, taking care not to burn the mixture.
Whisk the butter and syrup into the confectioner's sugar until the glaze is very smooth. Use it immediately if you want to drizzle it over the banana bread. It thickens as it cools, and can also develop a skin if it sits for too long. If you want to allow it to thicken, that's fine, just be sure to stir it thoroughly before attempting to use it.
Note: if you don't have confectioner's (a/k/a powdered) sugar, you can make your own. I pretty much never have any on hand, but I always have plain white sugar in the cupboard, which you can simply pulverize in a blender to get confectioner's sugar! Regardless of whether you use the pre-pulverized stuff, or make it yourself, I recommend sifting it before adding it to the glaze to ensure you don't have any lumps!