Preheat the oven to 410°F/210°C. Grease a 9-inch/23 cm square cake pan.

Make the cake: In a bowl, mash the bananas with a potato masher or a fork. Set aside.

In a separate bowl, whisk together the flour, baking powder, and 1/2 teaspoon salt. Set aside.

In a stand mixer fitted with the paddle (or in a bowl with a handheld mixer), beat the butter on high speed until smooth and creamy, about 1 minute. Add the brown sugar and beat on high speed for 2 minutes until creamed together. Scrape down the sides and bottom of the bowl with a rubber spatula as needed. Add the egg yolks to the butter/sugar mixture one at a time, beating after each. Beat in the mashed bananas and rum, scraping down the sides of the bowl as needed.

Add the flour mixture and mix until just incorporated. Do not overmix. The batter will be slightly thick, with a few lumps.

In a separate bowl, whip the egg whites with a pinch of salt until they hold stiff peaks. Gently fold the beaten egg whites into the batter. Spread the batter into the prepared pan.

Bake for 10 minutes. Reduce the oven temperature to 320°F/160°C and bake until a toothpick inserted in the centre of the cake comes out clean, another 50 minutes.

Remove the cake from the oven and cool completely in the pan on a wire rack.

Make the sauce: In a small saucepan, bring the beer to a gentle boil over medium heat. Reduce the heat to medium-low and simmer, stirring occasionally, until the beer is reduced to about 1⁄4 cup/60 mL (about 10 minutes). Remove from the heat and allow to cool for a few minutes.

In a medium saucepan, combine the brown sugar, butter, and 3 tablespoons beer reduction over high heat, and boil for 2 minutes. Remove from the heat. Stir in the vanilla and heavy cream, return to the heat and bring to a boil again. Boil for 1 minute. Remove from the heat and allow to cool. (The caramel will thicken as it cools.)

Serve slices of cake with a drizzle of beer-caramel sauce.