Preheat the oven to 350°F. Grease a 9- x 5-inch loaf pan. (I use vegetable shortening for greasing.)
Beat together the mashed bananas, brown sugar, butter, egg, mayonnaise, rum, and vanilla in a large bowl with an electric mixer until smooth.
Mix together the flour, baking soda, salt, and cinnamon in a medium bowl. Add to the banana mixture and stir just until combined.
Pour banana mixture into the prepared pan. Top with the sliced banana, cut sides up. Bake the bread at 350°F for 1 hour and 5 minutes, or until a toothpick inserted in the center comes out clean.
Let the bread cool in the pan on a wire rack. While the bread cools, make the browned butter-rum glaze. Cook the butter in a small skillet over medium heat for 2 to 3 minutes, swirling the pan occasionally, until the milk solids begin to brown and smell nutty.
Remove the pan from the heat and carefully stir in the rum. Return to the heat and cook for 30 seconds to 1 minute longer to cook off the alcohol. (If using a gas burner, carefully flambé the mixture.) Transfer the glaze to a small bowl and whisk in the sugar until smooth.
Remove the bread from the pan and drizzle with the glaze.