Preheat the oven to 350°F (175°C). Spread the hazelnuts on a baking sheet and toast until fragrant and skins split, 10–12 minutes. Rub warm nuts in a clean towel to remove most skins, then let cool 5 minutes.
Grease a 9-inch (23 cm) springform pan with the softened butter and line the base with parchment paper.
Pulse the cooled hazelnuts in a food processor with 2 tbsp of the granulated sugar until very finely ground, stopping before they turn pasty, 20–30 seconds total.
Melt the 6 tbsp unsalted butter and let cool slightly. In a large bowl, whisk the egg yolks with half of the remaining sugar until pale and thick, 2–3 minutes. Whisk in the melted butter and lemon zest.
In a separate clean bowl, beat the egg whites with the fine salt to soft peaks. Gradually sprinkle in the remaining sugar and continue beating to glossy medium peaks, 2–3 minutes.
Fold the ground hazelnuts and baking powder into the yolk mixture until evenly combined; the batter will be thick.
Fold in one-third of the beaten egg whites to loosen the batter, then gently fold in the remaining whites just until no white streaks remain.
Scrape the batter into the prepared pan and smooth the top. Bake until risen, golden, and a toothpick inserted in the center comes out with a few moist crumbs, 30–35 minutes.
Cool in the pan for 10 minutes, then release the ring and cool completely on a wire rack, 45–60 minutes.
Dust with powdered sugar. Slice and serve plain or with whipped cream.