Heat the oven to 170 C / 340 F. Butter a 23 cm / 9-inch springform pan, line the base with parchment, and dust the sides with flour.
If your hazelnuts aren't pre-toasted, spread them on a sheet pan and toast at 175 C / 350 F for 10 to 12 minutes until the skins crack and they smell toasted. Rub the warm nuts in a clean towel to remove the skins.
Pulse the cooled hazelnuts with 2 tbsp of the sugar in a food processor until finely ground, stopping before the mixture turns to paste.
Beat the egg yolks with the remaining sugar in a large bowl until pale and thick, about 3 minutes.
Beat in the melted butter, ground hazelnuts, flour, baking powder, and lemon zest just until combined.
In a separate clean bowl, whip the egg whites with the pinch of salt to soft peaks.
Fold the whites into the batter in three additions, cutting gently through the mixture rather than stirring, to keep the air in.
Pour the batter into the prepared pan and smooth the top. Bake for 35 to 40 minutes, until a toothpick comes out with a few moist crumbs and the edges pull away from the pan sides.
Cool in the pan on a rack for 15 to 20 minutes, then release the springform and cool the cake completely on the rack.
Dust with powdered sugar just before serving.