Preheat the oven to 350 degrees.
Whisk together the milk, sugar, rum and cornstarch in a large bowl.
Add the cubes of bread and mix well. Set the bread aside to soak.
Prepare a large baking dish by spraying with cooking spray.
Arrange half the bananas in the bottom of the baking dish. Scoop in half the bread mixture. Then layer on the remaining bananas, and add the remaining bread on top.
Bake in the preheated oven for 35 minutes or until the pudding has thickened. Turn on the broiler and let the bread brown a little on top, for another 3-4 minutes. Keep a close eye on the pudding while the broil function is on because you don't want it to over-brown.
Mix all ingredients in a small saucepan. Bring to a boil, whisking constantly. Turn off the heat as the sauce starts to thicken.
Pour the sauce over the pudding and serve with ice cream or by itself.
Here's the recipe for my Vegan Olive Oil Brioche with Aquafaba. Works perfectly in this pudding.
If you'd rather not add the sauce to the recipe-- and the pudding is delicious without it-- you can scatter on some golden raisins.