Chocolate pudding cake

TOTAL TIME: Prep: 1-1/4 hours + chilling Bake: 35 min. + cooling MAKES: 8 servings
Ingredients
PUDDING:
1/2 cup sugar
2 tablespoons cornstarch
1/4 teaspoon salt
1-1/2 cups whole milk
3 ounces semisweet chocolate, chopped
1 teaspoon Pure Vanilla Extract
CAKE:
1-1/2 cups all-purpose flour
2 teaspoons baking powder
1/2 teaspoon baking soda
1/2 teaspoon salt
1/2 cup unsalted butter, cubed
3/4 cup Dutch-processed cocoa
1 cup sugar
1 cup packed brown sugar
1 cup buttermilk
1 cup strong brewed coffee
1 teaspoon Pure Vanilla Extract
2 eggs
FROSTING:
8 ounces semisweet chocolate, chopped
1/2 cup unsalted butter, cubed
1/3 cup hot water
2 teaspoons light corn syrup
2 teaspoons Pure Vanilla Extract