Preheat the oven to 350 degrees F.
Lightly grease a 9 by 13-inch baking dish with 1 tablespoon of the butter and set aside.
Place the cubed bread in a large mixing bowl. In a small saucepan, melt 4 tablespoons of the butter over medium heat and pour the butter over the bread cubes. Using a rubber spatula, toss to evenly distribute the butter.
Combine the cream, milk, eggs, brown sugar, granulated sugar, vanilla, cinnamon, nutmeg, salt, and banana liqueur in a medium mixing bowl, whisking to combine. Pour the cream mixture over the bread cubes and stir to combine. Let sit for 30 for 45 minutes, or until the bread is soft and has absorbed most of the liquid.
Stir in the mashed bananas and walnuts. Pour the mixture into prepared dish and bake, uncovered, for 45 to 50 minutes, until the top is golden brown and crispy and the pudding is firm in the center. Prepare the Buttery Rum Sauce while the pudding is baking.
Remove the pudding from the oven and cool on a wire rack for 15 minutes before serving. Serve warm with Buttery Rum Sauce and whipped cream, if desired.
In a heavy, nonreactive saucepan combine all ingredients except the rum and cook over medium-low heat, whisking constantly, until the sauce is thickened and coats the back of a spoon. Remove from the heat immediately; do not allow the sauce to boil. Add the rum, stir to combine, and allow to cool slightly. Serve warm with Banana Walnut Bread Pudding. (The sauce can be made up to 1 week in advance and stored, refrigerated, in an airtight container. Warm gently before serving.)