In a medium bowl, stir together the Oreo cookie crumbs and melted butter. Evenly divide the crumbs between your individual serving dishes and press into the bottoms of the dishes to form a crust layer.
In a large bowl, with an electric mixer, beat the cream cheese and Nutella until smooth. Add vanilla and mix to combine. Using a rubber spatula, fold in the whipped topping until well blended and no streaks remain.
Evenly pipe or spoon the filling into individual serving dishes. Cover with plastic wrap and refrigerate for at least 2 hours before serving.
If desired, garnish with additional whipped topping, chocolate shavings, and/or toasted, chopped hazelnuts.