Pre-heat the oven to 350°F.
In a small batter bowl, microwave 3 tablespoons of the butter and all of the brown sugar, uncovered, on high for 30 to 60 seconds or until the butter is melted. Remove from the microwave and add the rum; whisk until smooth; set aside.
In a large non-stick skillet (cast iron preferred), over medium heat, melt the remaining butter. Slice 4 of the bananas in half widthwise; then quarter each half lengthwise.
Add the sliced bananas, cut-side down, into the pan; allow them to cook undisturbed for 4 to 5 minutes, or until the bananas are lightly caramelized; remove the pan from the heat.
In a separate small bowl, mash the remaining banana; add the eggs, water and vanilla extract; whisk to combine.
Add the cake mix and whisk until smooth.
Whisk the brown sugar mixture and pour evenly over the bananas.
Carefully pour the cake batter evenly into pan, starting from outside and working your way in.
Place in the pre-heated oven and bake for 22 to 24 minutes, or until a cake tester, inserted in the center, comes out clean. Remove from the oven and gently release the sides of the cake with the a knife.
Allow the cake to stand for 2 to 3 minutes; invert the pan onto a serving platter to remove from skillet.