With an electric mixer (or a milk frother that can whip cream), beat 1/2 cup heavy cream and powdered sugar until medium to stiff peaks form. This is the topping for the Viennese coffee. Refrigerate until ready to use.
In a saucepan or microwave-safe bowl, heat the remaining 1/2 cup of cream over medium heat until steaming, stirring occasionally. Remove from the heat and whisk in chopped chocolate (or chocolate chips).
Slowly whisk the cream into the strong brewed coffee or espresso. Divide evenly among cups, top with whipped cream and additional chocolate shavings.