Preheat the oven to 325°F (160°C) and line an 8×8 inch baking dish with parchment paper.
Separate the eggs. In the bowl of a stand mixer or with an electric mixer, beat the egg whites until stiff peaks form. Transfer to a clean bowl and set aside.
In the mixer bowl, beat the egg yolks with 1 cup icing sugar until pale and creamy.
Add the Nutella, melted cooled butter, vanilla extract, salt, and cocoa powder to the yolk mixture and beat until combined.
On low speed, add the flour one tablespoon at a time until incorporated, then slowly stream in the lukewarm milk while mixing until the batter is very thin and smooth.
Gently fold one-third of the beaten egg whites into the batter to loosen it, then fold in the remaining egg whites in two additions until mostly combined; small streaks of foam are fine.
Pour the batter into the prepared dish and bake for 40–70 minutes, or until the top is browned and the center is set and no longer wobbly.
Cool the cake completely, then refrigerate for at least 2 hours to finish setting.
Just before serving, dust the top with icing sugar and slice.