In a large bowl, whisk pumpkin, eggs, half and half, brown sugar, spices, salt and vanilla. Stir in bread cubes and let stand 15 minutes.
Pour bread misture into a greased 8x8-inch square glass baking dish and bake at 350F for 20-25 minutes, or until custard is set.
TO MAKE THE CARAMEL RUM SAUCE: Combine all ingredients in a medium saucepan. Bring to a boil over medium heat, stirring occasionally. Turn burner to medium low and let sauce bubble for 3-4 minutes. Serve warm drizzled over the bread pudding.