Preheat the oven to 350 degrees F. Grease an 8-by-4-inch loaf pan with butter and line the bottom and 2 sides with parchment paper.
In a medium bowl, combine the flour, baking powder, baking soda and salt. In a large bowl, cream together the granulated sugar and eggs. Stir in the mashed bananas, oil, cinnamon and vanilla. Stir in the flour mixture a third at a time and mix until just combined. Fold in the sliced bananas.
Pour the batter into the prepared pan. Bake until a skewer inserted in the center comes out mostly clean (some moist crumbs should stick to it), about 1 hour 15 minutes. Let cool completely.
Meanwhile, combine the butter, brown sugar and 2 tablespoons water in a saucepan and bring to a boil, stirring to dissolve the sugar. Remove from the heat and stir in the rum.
While the bread is cooling, poke it all over with a skewer. Brush with half of the rum sauce and allow it to sit and soak through. Remove the banana bread from the loaf pan once cool, slice and serve with a scoop of ice cream and a drizzle of the remaining rum sauce.
If you don't have an 8-by-4-inch pan, a 9-by-5-inch pan will work; just cut the baking time by 10 minutes.
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