To prepare pudding, combine first 9 ingredients in a large bowl, stirring with a whisk. Add bread; toss gently to combine. Cover and chill for 1 hour.
Preheat oven to 350°.
Spoon bread mixture into an 11 x 7-inch baking dish coated with cooking spray. Bake bread mixture at 350° for 50 minutes or until set.
To prepare sauce, combine 1/3 cup granulated sugar and butter in a medium saucepan over medium heat. Combine 1 cup milk and cornstarch in a small bowl, stirring with a whisk. Add milk mixture to pan; bring to a boil. Cook 4 minutes or until thick, stirring constantly. Reduce heat to medium-low; stir in 2 tablespoons bourbon. Cook 2 minutes; serve over pudding.