Preheat oven to 350°F (175°C). Grease a 9x13-inch baking dish with butter or cooking spray.
Spread bread cubes on a baking sheet and toast in the oven for 8-10 minutes until slightly dried out but not browned. Transfer to the prepared baking dish.
In a large mixing bowl, whisk together the milk, heavy cream, eggs, sugar, cocoa powder, vanilla extract, and salt until smooth and well combined.
Warm the Nutella in the microwave for 15-20 seconds until slightly softened. Add to the custard mixture and whisk until mostly incorporated with some swirls remaining.
Pour the chocolate custard over the bread cubes, pressing down gently with a spatula to ensure all pieces are soaked. Let stand for 15 minutes so the bread absorbs the liquid.
Scatter chocolate chips and chopped hazelnuts over the top, then drizzle with melted butter.
Cover the dish with aluminum foil and bake for 30 minutes. Remove foil and continue baking for 20-25 minutes until the top is golden and the center is set but slightly jiggly.
Let cool for 10-15 minutes before serving. Drizzle with additional warmed Nutella if desired and serve warm with whipped cream or vanilla ice cream.
