Preheat your oven to 350°F (175°C) and grease a 9x13-inch baking dish with unsalted butter.

In a large mixing bowl, combine the cubed day-old bread with the sliced ripe bananas and gently toss them together.

In a separate bowl, whisk together the eggs, milk, heavy cream, granulated sugar, vanilla extract, ground cinnamon, and salt until fully blended.

Pour the custard mixture over the bread and bananas, gently folding everything together. Allow to sit for about 15 minutes.

Melt the unsalted butter in a small saucepan over medium heat. Stir in the brown sugar and dark rum, cooking until the mixture is smooth and bubbling.

If using, fold in the chopped walnuts into the bread mixture before transferring it to the prepared baking dish.

Pour the bread mixture into your greased baking dish and drizzle the warm rum topping evenly over the top.

Bake for 40-45 minutes, until the top is golden brown and a knife inserted in the center comes out clean.

Once baked, remove the dish from the oven and let it cool for about 15 minutes before serving.

Spoon out the warm bread pudding into bowls and drizzle any remaining sauce on top if desired.