Coat a 9×13 baking dish with nonstick cooking spray. Arrange the bread cubes and raisins evenly in the prepared baking dish.
In a large bowl, whisk the eggs half-and-half together. Add rum, sugar, vanilla, cinnamon and salt and whisk to combine.
Pour the custard mixture evenly over the bread mixture. Cover with plastic wrap and weigh down with another baking dish. Refrigerate at least one hour, up to overnight.
Preheat oven to 350 degrees. Uncover the bread pudding. Melt the butter and pour evenly over the casserole. Bake 60-65 minutes, until center is set. Let sit for 30 minutes before serving.
While the bread pudding is cooling, make the buttered rum sauce: Combine sugar and cornstarch in a small bowl. Melt butter over low heat in a small saucepan. Stir in the cornstarch mixture. Pour in the milk. Bring to a boil over medium-high heat, stirring. Reduce heat to low and stir until thickened. Off heat, stir in the rum. Pour warm sauce over bread pudding.