Generously butter a 12-cup ovenproof baking dish.
To make the custard, whisk the milk, cream, and sugar together in a large bowl. Add the egg yolks and the whole egg one at a time, whisking well after each addition. Whisk in the vanilla, nutmeg, and salt, and set aside.
Cut four slices of the banana bread vertically into 1-inch-wide fingers. Arrange an even layer of bread fingers on the bottom of the prepared dish, cutting and fitting them to cover the bottom. It is fine if they curve slightly up the sides of the dish. Cover the bread slices with an even layer of banana slices, and then pour enough custard over the bread and bananas to cover them. Repeat the layers of bread and bananas. Cut the remaining slice of banana bread into cubs, and sprinkle these over the bananas, press down slightly on the bread cubes, and cover. Set aside for 30 minutes.
Meanwhile, preheat the oven to 350F.
Bake the pudding until it is almost solid through, about 45 minutes. Then remove the cover and bake until the cubes of banana bread on the top are golden, an additional 15 minutes. Remove the dish from the oven and cool it on a wire rack.
While the bread pudding is cooling, make the sauce: Melt the butter in a small saucepan over medium heat. Whisk in the brown sugar, lemon juice, rum, and water, and bring to a rolling boil. Cook, whisking occasionally, until the mixture begins to thicken and the sugar has dissolved, about 5 minutes. Remove the sauce from the heat and transfer it to a serving pitcher. Serve along with the slighly cooled banana bread pudding and whipped cream, if you like.
Source: "The Farm House Cookbook," Susan Herrmann Loomis, 1991.