Preheat the oven to 350°. Lightly butter eight 4 1/2-by-2-inch ramekins. In a large bowl, toss the brioche with the melted butter. Spread the brioche on a large baking sheet and bake for about 15 minutes, stirring occasionally, until golden and crisp.
Meanwhile, in a small heavy saucepan, combine 3/4 cup of the sugar with the corn syrup and water; cook over moderate heat, stirring until the sugar dissolves. Boil the syrup without stirring until a deep amber caramel forms, about 7 minutes. Remove the caramel from the heat and carefully stir in 1/3 cup of the cream, the remaining 3 tablespoons of butter and the salt. Immediately pour 2 tablespoons of the caramel into each of the prepared ramekins and set them in a large roasting pan. Arrange the sliced bananas decoratively in the caramel.
In a large bowl, mash the remaining 2 bananas until smooth. Whisk in the milk, egg yolks, the remaining 1/2 cup of sugar, the vanilla and the remaining 1 cup of heavy cream until smooth. Fold in the toasted brioche and let stand, stirring occasionally, until the liquid is completely absorbed, about 20 minutes.
Spoon the bread mixture into the ramekins and smooth the surfaces. Pour enough hot water into the roasting pan to reach halfway up the sides of the ramekins. Bake the puddings in the water bath for 35 minutes, or until firm and a knife inserted in the center comes out clean. Remove the ramekins from the water bath and let cool slightly.
To unmold, run a knife around the puddings. Top each with a plate. Invert the plates and lift off the ramekins. Pour any remaining caramel around the puddings. Serve warm with ice cream or whipped cream.
The baked puddings can be wrapped tightly with foil and refrigerated overnight. Rewarm, covered, in a 325° oven for about 15 minutes before unmolding.
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