This cake, with its subtle toasted-nut flavor and moist, buttery texture, is one of the few things I can't resist, especially when it's still warm from the oven. It's also delicious made with walnuts or almonds in place of the hazelnuts.
In a saucepan, melt the butter over medium heat, and continue to cook it until it turns brown and smells quite nutty, being careful not to let it burn, 10 to 15 minutes. Remove from the heat and let cool almost to room temperature.
Meanwhile, heat a convection oven to 325 degrees F or a conventional oven to 350 degrees F. Butter the bottom of a 10-inch round cake pan and line the bottom of the pan with parchment paper.
Spread the hazelnuts on a rimmed baking sheet and toast them in the oven until lightly browned and fragrant, 10 to 15 minutes. Wrap the hazelnuts in a clean kitchen towel and rub them together vigorously to remove most of the skins. (It's fine if some is left on.)
Chop about 1/4 cup of the nuts finely. Grind the rest of the toasted nuts in a food processor until finely ground.
Sift the flour, confectioners' sugar, and salt together into a large bowl. Add the ground nuts and whisk to combine. Add the egg whites and whisk them together well with the dry ingredients. Slowly add the melted butter, whisking constantly until it is completely incorporated and no butter is floating on top of the mixture.
Pour the batter into the prepared pan and sprinkle the top with the reserved chopped nuts. Bake until the cake feels solid but still gives slightly when touched, 40 to 45 minutes. Let the cake cool in the pan for 10 minutes before inverting it onto a serving plate.