Preheat the oven to 350 degrees F. Butter just the sides of a 9-inch round cake pan.
For the caramel topping: Stir the sugar and 1/3 cup water in a medium saucepan over medium heat until the sugar dissolves. Stop stirring, and gently swirl the saucepan occasionally until the mixture is the color of deep amber, 8 to 9 minutes. If needed, brush any sugar crystals off the side of the pan with a wet pastry brush. Remove the saucepan from the heat, and carefully whisk in the butter (take care: the mixture can bubble up considerably). Pour the caramel into the prepared cake pan, tilting the pan to cover most of the bottom (be careful: the pan will be very hot from the caramel). Arrange the bananas in a single, tight layer, then cover any gaps with the remaining slices.
For the cake: Whisk together the flour, baking powder, baking soda and salt in a medium bowl. Whisk together the milk and sour cream in a small bowl.
Beat the butter and sugar together with an electric mixer on medium-high speed in a large bowl until light and fluffy, about 4 minutes. Add the eggs 1 at a time, beating after each addition and scraping down the side of the bowl as needed, then beat in the vanilla. Reduce the mixer speed to low, and add the flour mixture in 3 batches, alternating with the milk-sour cream mixture, mixing until just incorporated.
Pour the batter over the bananas in the cake pan, and spread evenly. Bake the cake until it is golden brown and bounces back when pressed and a toothpick or cake tester comes out clean when inserted in the center, 50 to 60 minutes. Run a knife around the edge of the cake to separate it from the pan. Poke the cake all over with a skewer, and brush the cake with the rum. Let cool in the pan on a rack until just cool enough to handle, 20 to 30 minutes. Invert a serving platter over the cake pan and, while holding them together, quickly flip them over so the cake unmolds, banana-side up. Let cool completely. Cut into wedges, and serve.
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