Place milk in a medium saucepan and bring to a simmer. Whisk in both chocolates and heat until melted and smooth. Remove from the heat.

Whisk together the yolks, sugar, and cornstarch in a large bowl. Slowly whisk in the chocolate-milk mixture and then return the mixture to the saucepan over medium-high heat and whisk until thickened. Remove from the heat, stir in the vanilla, and transfer to a clean bowl. Place a piece of plastic wrap over the surface of the pastry cream and let cool to room temperature.

Spoon about 1 cup of the pastry cream into a 9 by 13-inch glass dish. Layer the cookies, bananas, and pastry cream in the bowl, ending with the pastry cream. Cover tightly and refrigerate until chilled, at least 4 hours or overnight. Just before serving, spread the whipped cream over the top of the pudding and sprinkle with the chocolate shavings.