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1
Generously butter a 12-cup ovenproof baking dish.
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2
To make the custard, whisk the milk, cream, and sugar together in a large bowl.
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3
Add the egg yolks and the whole egg one at a time, whisking well after each addition.
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4
Whisk in the vanilla, nutmeg, and salt, and set aside.
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5
Cut four slices of the banana bread vertically into 1-inch-wide fingers.
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6
Arrange an even layer of bread fingers on the bottom of the prepared dish, cutting and fitting them to cover the bottom.
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7
It is fine if they curve slightly up the sides of the dish.
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8
Cover the bread slices with an even layer of banana slices, and then pour enough custard over the bread and bananas to cover them.
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9
Repeat the layers of bread and bananas.
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10
Cut the remaining slice of banana bread into cubs, and sprinkle these over the bananas, press down slightly on the bread cubes, and cover.
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11
Set aside for 30 minutes.
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12
Meanwhile, preheat the oven to 350F (180C).
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13
Bake the pudding until it is almost solid through, about 45 minutes.
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14
Then remove the cover and bake until the cubes of banana bread on the top are golden, an additional 15 minutes.
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15
Remove the dish from the oven and cool it on a wire rack.
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16
While the bread pudding is cooling, make the sauce: Melt the butter in a small saucepan over medium heat.
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17
Whisk in the brown sugar, lemon juice, rum, and water, and bring to a rolling boil.
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18
Cook, whisking occasionally, until the mixture begins to thicken and the sugar has dissolved, about 5 minutes.
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19
Remove the sauce from the heat and transfer it to a serving pitcher.
