Preheat the oven to 350˚F. Grease a 9-inch round cake pan with nonstick spray and line with parchment. Place the bananas cut side down on the parchment, arranging or trimming to fit.
In a medium saucepan, combine the sugar and water over medium-high heat. Cook, shaking the pan if needed but not stirring, until an amber caramel forms, about 6 minutes. Immediately pour over the bananas in an even layer.
In a large bowl, whisk the brown sugar and mashed bananas until a relatively smooth paste forms. Whisk in the olive oil, yogurt, vanilla, and eggs until well incorporated.
Add the flour, salt, cinnamon, baking soda, and baking powder. Stir until just incorporated and there are no pockets of flour, then fold in the chocolate.
Pour the batter over the caramel-coated bananas and smooth with a rubber spatula. Bake for 55 to 60 minutes, until golden and cracked with an internal temp of 190°F.
Let cool for 15 minutes, then run a paring knife along the edges of the cake. Place a plate over the cake pan and invert, discarding parchment paper. Let cool slightly, then serve warm.
