1. Preheat oven to 350°F. Spray a 9×5 loaf pan with nonstick spray. Line the bottom and up the short sides with a strip of parchment paper. Spray again with nonstick spray. Set aside.
2. In a medium saucepan over medium-low heat melt the butter. Bring it to a boil, stirring frequently, until the butter turns an amber color, about 5 minutes. Remove from the heat and allow to cool for 20 minutes.
3. When butter has cooled for 20 minutes, mix in the light brown sugar and rum. Stir until smooth, making sure to scrape up the brown bits at the bottom of the saucepan. Set aside.
4. In a medium bowl mash 2 of the bananas with a fork, leaving them slightly lumpy. Slice the remaining bananas thinly, and set aside.
5. Add the mashed bananas, eggs, and sour cream into the sugar mixture, and stir to combine.
6. In a large bowl whisk together the flour, cinnamon, baking soda, and salt. Mix in the wet ingredients until just combined. Stir in the walnuts and sliced bananas until evenly incorporated.
7. Pour batter into the prepared pan and bake for 50-60 minutes, or until a toothpick inserted in the center comes out clean.
8. Cool in the pan for 15 minutes, and then using the parchment paper as handles, carefully lift the banana bread out of the pan to cool on a wire rack.

Icing

1. In a medium bowl whisk together the powdered sugar, rum and milk until smooth. Drizzle on top of the bread.
2. Serve warm or at room temperature.