In a medium saucepan, bring the toffee sauce ingredients to a boil over medium heat, whisking often.
Reduce the heat to medium-low, and let the sauce gently simmer for about 15 minutes, until it’s reduced to about 1-½ cups. Be sure to keep stirring the sauce to avoid it sticking to and burning on the bottom of the pan.
The sauce is ready when it coats the back of a wooden spoon. When the pan is off the heat, add the rum.
You can either use it right away or store it in the fridge for the next day. Just reheat it in the microwave for a minute or so.
For the cake
Preheat the oven to 350 °F (175 °C). Grease and flour an 8×8-inch (20x20 cm) square, metal baking pan (a brownie pan). Be sure to tap out the excess flour.
Whisk together the flour and baking powder in a small bowl. Set aside for later.
In the bowl of an electric mixer fitted with the paddle attachment, cream the butter and the brown sugar until smooth.
Add the eggs, one at a time, mixing well with each addition, and scraping down the bowl as needed.
Add the mashed banana, rum, and vanilla, and mix well.
With the mixer on low, add the dry ingredients slowly, in 4 additions, scraping down the bowl between additions. Mix the batter until the flour is combined.
Spread the batter in the prepared pan, and bake the cake for 35–38 minutes (or until a tester comes out clean).
Remove the pan from the oven and place on a cooling rack. Pour half a cup of the toffee sauce over the hot cake, and return the pan to the oven.
Continue baking for 6 minutes. The sauce will bubble,
Remove the pan from the oven, and let the cake cool slightly (if serving warm), or completely (if serving later) in the pan.
To serve
Cut the cake into 9 servings.
Serve each piece of cake with several spoonfuls of warm toffee sauce, a couple slices of banana, and a scoop of ice cream. Pass around the toffee sauce because your guests will want more.