Step 1In a large, heavy-bottom saucepan, melt the butter over medium heat. Stir in the brown sugar and cook, stirring to dissolve. Stir in the cream, vanilla and a pinch of salt and gently bring the mixture to a boil, stirring frequently. When the mixture has come to a boil, remove from heat and set aside to cool.
Step 2While the caramel cream cools, toss the bananas and croissants in a large mixing bowl.
Step 3Whisk the eggs into the caramel cream to form a custard base, strain and then pour the custard base over the croissants and gently toss. Place the mixture in a greased 9-inch round cake pan or an 8-inch square baking dish. Set the mixture aside to soak for 1 hour. Meanwhile, heat the oven to 350 degrees.
Step 4Place the cake pan into a larger baking dish; fill the larger baking dish with enough boiling water to come up the side of the cake pan halfway to form a water bath.
Step 5Bake the bread pudding until puffed and set, about 45 minutes. Remove from the oven and water bath, and cool slightly before serving. For a crisp topping, place the cake pan under the broiler just until it begins to brown and crisp on top, about 1 to 3 minutes, depending on the broiler.