Start with the Rum-soaked bananas: In a small bowl, mix the bananas slices with all the other ingredients and leave to soak. Make sure the bananas are covered with the rum. Cover the bowl and set aside.
Preheat the oven to 350F/175C. Line a 9x13-inch baking pan with parchment paper.
In a large heat-proof bowl, put the chocolate and butter and melt in the microwave in 20 second-intervals, stirring in between each interval, until melted (or set it over a saucepan of simmering water, stirring occasionally). Add the sugar and whisk until combined.
Add the eggs, one at a time, and whisk just until combined. Add the vanilla and stir. Do not over mix.
Sift the flour and salt over the chocolate mixture and fold, using a rubber spatula, just until mixed. Fold in the bananas with any left liquid from the bowl.
Pour the batter into the prepared pan and smooth the top. Bake for about 25-30 minutes until a toothpick inserted into the center of the brownies comes out with a few moist crumbs. Let the brownies cool completely, then lift them out of the pan using the parchment paper and cut into squares.