Preheat the oven to 350°F (175°C) and grease a baking dish.

Cut the day-old bread into bite-sized cubes and spread evenly in the baking dish.

In a large mixing bowl, whisk together the eggs, milk, heavy cream, brown sugar, vanilla extract, and ground cinnamon until smooth.

Mash the ripe bananas until smooth and mix into the custard until fully incorporated.

Pour the custard mixture over the bread cubes, ensuring each piece is well-coated.

Let the combined dish sit for about 10 minutes to absorb the custard.

Bake in the preheated oven for 40-45 minutes until the top is golden brown and the center is slightly jiggly.

Warm the caramel sauce in a small saucepan over low heat until smooth.

Once baked, let the pudding cool for a few minutes and drizzle the warm caramel sauce over the top before serving.