Preheat oven to 325°F (160°C). Line an 8×8-inch baking dish with parchment paper and set aside.

Separate the 4 eggs. Place the egg whites in the bowl of a stand mixer (or a large clean bowl if using a hand mixer).

Beat the egg whites on medium-high until stiff peaks form (peaks that stand straight up). Transfer the beaten egg whites to a clean bowl and set aside.

In the mixer bowl (wipe clean if needed), add the 4 egg yolks and 1 cup icing sugar. Beat until the mixture is pale yellow and slightly thickened, about 1–3 minutes.

Add 1 cup Nutella, ½ cup melted unsalted butter (room temperature), 1 teaspoon vanilla extract, ⅛ teaspoon salt, and 2 tablespoons cocoa powder to the yolk mixture. Mix on low to medium until the ingredients are fully incorporated and smooth, about 1 minute. Scrape the bowl once.

With the mixer on low, add the ¾ cup all-purpose flour (minus 2 tbsp) one tablespoon at a time, mixing just until each addition is incorporated. Scrape the bowl once or twice.

With the mixer still on low, slowly add the 2 cups lukewarm milk in a steady stream, pausing to scrape the bowl as needed, and mix until the batter is smooth and very thin (almost like crepe batter).

Using a spatula, gently fold one-third of the beaten egg whites into the batter to lighten it. Then fold in the remaining egg whites in two additions, folding just until mostly combined. Small foamy streaks are okay.

Pour the batter into the prepared 8×8-inch pan and smooth the top with a spatula.

Bake on the center rack for 40–70 minutes, beginning to check at 40 minutes. The cake is done when the top is browned, the surface is firm to the touch, and the cake does not jiggle in the center. (If using a toothpick, the top should be set; some moistness below is expected.)

Remove the cake from the oven and let it cool completely at room temperature in the pan.

Once cooled, refrigerate the cake in the pan for at least 2 hours to set.

Just before serving, dust the top with additional icing sugar. Slice and serve chilled or slightly chilled.