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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.
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Preheat the oven to 350F/175C. Line a 9x13-inch baking pan with parchment paper.
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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.
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Add the eggs, one at a time, and whisk just until combined. Add the vanilla and stir. Do not over mix.
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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.
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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.