Preheat the oven to 350°F Grease an 8-inch square baking dish with unsalted butter.
Make 6 sandwiches with the bread and nutella. Cut each sandwich into 6 to 8 pieces. Pile the sandwich pieces into the prepared baking dish.
In a small saucepan, combine the cream, salt, and vanilla and cook over medium heat until warm.
In a large bowl, whisk together the sugar and eggs. Slowly add the cream mixture, stirring constantly to make sure the eggs do not cook.
Pour the mixture over the bread and cover the dish with aluminum foil.
Bake on the center rack of the oven for 30 to 40 minutes, removing the foil after about 20 minutes. The finished pudding will be golden brown on top.
Allow the pudding to rest for at least ten minutes before serving.