Soak the raisins in the rum for an hour before starting to prepare the bread pudding. They can alternatively be soaked overnight too.
Preheat your oven to 350 degrees F.
Place the pieces of bread in a greased 9 x 13 baking pan. Drizzle the bread with the melted butter, and then pour the rum and raisins over the bread. Make sure that the raisins are evenly distributed over the bread.
In a large bowl, whisk together the milk, brown sugar, white sugar, eggs, cinnamon, and vanilla.
Pour the mixture over the bread, and lightly push down the bread with a fork until all the bread pieces are covered and soaking up the liquid.
Bake in the preheated oven for 35 – 45 minutes or until the top is golden brown and it springs back when lightly pressed. The bread pudding will puff up and then fall, don’t worry, that is what it is supposed to do.
While the bread pudding is baking mix the cornstarch and sugar together in a small bowl and set aside.
Melt the butter in a small saucepan. Reduce the heat to low, then add the heavy cream and the sugar cornstarch mixture. Stir over low heat until the mixture begins to thicken. Be careful not to allow the mixture to boil, or it will burn.
Remove from the heat and add 2 tbsp of amber rum. Taste, and if desired, add an additional tbsp of rum.
Once the bread pudding is baked, serve the rum sauce over the top of individual servings.