Bring 2 cups whipping cream, liqueur, sugar, and vanilla to boil in heavy medium saucepan over medium-high heat, stirring frequently. Mix cornstarch and 2 teaspoons water in small bowl to blend; whisk into cream mixture.
Boil until sauce thickens, stirring constantly, about 3 minutes. Cool, then cover and refrigerate until cold, about 2 hours. (Can be made 3 days ahead. Keep refrigerated.)
Make the bread pudding:
Preheat oven to 350°F. Spray 13x9x2-inch glass baking dish with nonstick spray.
Combine bread, chocolate, and white chocolate in large bowl; toss to blend.
Using electric mixer, beat eggs, ½ cup sugar, vanilla, and milk in another large bowl to blend.
Gradually beat in 1 ½ cups heavy whipping cream. Add cream mixture to bread mixture; stir to combine. Let stand 30 minutes.
Transfer bread mixture to prepared dish, spreading evenly. Drizzle with remaining 1/2 cup heavy whipping cream. Sprinkle with remaining 2 tablespoons sugar.
Bake pudding until edges are golden and custard is set in center, about 1 hour. Cool pudding slightly.
Warm up the sauce and drizzle the bread pudding with the sauce and serve warm.