Preheat the oven to 350F (175C). Line the bottom and sides of an 8-inch (20 cm) springform pan with parchment paper.

Arrange the bananas, cut side down, on the bottom of the pan.

Place the sugar and water into a heavy-bottomed saucepan. Heat the mixture over medium-high heat until it reaches a deep amber color. Pour in the hot heavy cream and whisk vigorously until completely incorporated. Remove from the heat and whisk in the salt. Pour the caramel over the bananas.

In a mixing bowl, add the butter, sugar, oil, and vanilla extract. Beat with a hand mixer until pale yellow and fluffy.

Add the egg and beat until well incorporated.

Add the pureed banana and beat again.

Sift in the flour, baking powder, and salt. Mix just until no dry ingredients remain.

Spread the cake batter evenly on top of the bananas. Bake for 40-45 minutes, or until the cake has risen and is golden brown on top. A toothpick inserted in the middle should come out clean.

Remove the cake from the oven and allow it to rest for 10 minutes.

Remove the springform ring and peel away the parchment from the sides. Place a serving plate over the cake, then invert it in one quick motion. Lift off the bottom of the pan and gently peel away the parchment.

Allow to cool completely before serving, or serve slightly warm with a scoop of vanilla ice cream.