Preheat oven to 350°F with rack in middle of oven. Whisk together cream, pumpkin, milk, syrup, eggs, salt and spices in a bowl.
Toss bread cubes with melted butter in another bowl, then add pumpkin mixture and toss to coat. Transfer to an ungreased 8-inch square baking dish and bake until custard is set, 40 to 45 minutes.
Melt butter in a small saucepan over medium heat. Stir in brown sugar and corn syrup until smooth. Bring to boil, stirring frequently; boil 1 minute. Remove from heat and stir in heavy cream, bourbon and vanilla. Serve warm with bread pudding.