Melt 1 tablespoon of the sugar in a heavy-bottomed saucepan on medium- high heat. While stirring constantly, continue to add sugar 1 tablespoon at a time until all is melted. Continue to cook until sugar turns light amber. Remove from the heat and carefully pour caramel into a 5-inch x 9-inch loaf pan to cover bottom evenly. Set aside and allow to cool.
Melt 1 tablespoon of the sugar in a heavy-bottomed saucepan on medium-high heat. While stirring constantly, add 1 tablespoon of sugar at a time until all is melted. Continue to cook until it becomes dark amber. Remove from the heat and carefully whisk in cream. Return to the heat and cook for another 2 minutes. Remove from the heat and add rum and salt. Strain through a fine-mesh sieve and allow to cool.
Place milk, vanilla bean and cinnamon stick in a saucepan on medium heat. Bring to a boil. Remove from the heat and set aside.
In a stainless steel bowl, whisk together eggs and sugar until light and fluffy. Temper egg mixture by whisking in 1/2 cup of the warm milk mixture. Whisk in the remainder of the warm milk, and transfer vanilla bean and cinnamon stick into mixture to infuse the flavours. Allow to cool and strain through a fine-mesh sieve. Set aside.
Preheat the oven to 350 F.
Butter the sides of the loaf pan that has the caramel in the bottom. Cut off and discard crusts from both loaves of French bread. Cut off a few 3/4-inch thick slices to line the bottom of the loaf pan and fit very tightly inside. Cut the remainder of bread into 1-inch cubes and set aside.
Dot the bottom layer of bread with 2 tablespoons of the butter. Sprinkle with ground cinnamon and nutmeg. Drizzle generously with caramel sauce. Peel banana and slice lengthwise to make long strips 1/4 inch thick. Lay a couple of banana slices on top of sliced bread. Pack in a layer of bread cubes, then pour 2 to 2 1/2 cups of custard over top. Gently press down on this layer to help bread to absorb custard. Dot with a little more of the butter. Repeat bread cubes, custard and butter until the loaf pan is full. Use your hands to press down gently on the top to ensure that all ingredients are incorporated together. The custard should not spill over the top but should be fully absorbed into the bread, giving it a spongelike feel and a dense texture.
Cut a piece of aluminum foil that is double the width of the loaf pan. Fold it in half and butter 1 side. Place the foil buttered side down and seal tightly to the loaf pan.
Place the loaf pan in a roasting pan. Pour hot water into the roasting pan until it reaches three quarters up the sides of the loaf pan. Bake in the oven for about 1 1/2 hours. When done, the banana bread pudding should be slightly puffed and be firm to the touch in the centre. Remove from oven and allow to rest for 10 to 15 minutes.
Run a small knife around the inside edges of the loaf pan and invert the pudding onto a flat surface. Gently tap the bottom of the loaf pan to break the suction that has been created during baking, to help the pudding to slide out easily. Place the pudding in an airtight container and refrigerate. Will keep for up to 4 days.
Cut pudding into 8 portions. Heat individual portions in the microwave for 30 seconds on high. Heat caramel sauce and pour 1/4 cup over each serving. Serve immediately.