Preheat oven to 350F. Arrange bread cubes in a lightly greased 13- x 9-inch pan.
Whisk together milk, pumpkin, eggs, maple syrup, bourbon, butter, sugar and pumpkin pie spice; pour over bread mixture, pressing bread cubes to absorb egg mixture.
Bake, covered, in preheated oven 30 minutes. Uncover and bake 30 more minutes or until lightly browned and set. Let stand 5 minutes before serving.
For the Bourbon sauce, melt 1/2 cup butter in a medium saucepan over low heat. Whisk in egg, sugar and vanilla; cook over low heat, stirring constantly 5 to 7 minutes or until mixture is thickened. Stir in bourbon and drizzle over Pumpkin Bread Pudding.