Prepare the caramelized bananas:
In a 8-inch skillet, melt the unsalted butter over low heat.
Sprinkle 1.5 tablespoons of sugar evenly over the melted butter.
Arrange the banana slices in a single layer on the sugar. Cook the mixture over very low heat for about 5 minutes, or until the sugar begins to caramelize and the bananas are slightly softened. Be careful not to burn the caramel.
Mix the cake batter:
In a bowl, whisk together the egg and 6 tablespoons of the sugar until light and fluffy.
Add the milk, salt, and vanilla extract to the egg mixture and stir until well combined.
Sift in the flour and baking powder, then stir until well combined, being careful not to overmix. Gradually pour in the cooking oil, stirring to gently incorporate it into the batter.
Bake the cake:
Carefully pour the cake batter over the caramelized bananas in the pan. Spread the batter evenly with a spatula.
Cover the pan tightly with a lid or foil to trap steam and set the heat to very low. Bake the cake on very low heat for about 25 minutes. Check occasionally to make sure it is baking evenly and not burning. Adjust the heat if necessary.
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