Preheat the oven to 350°. Generously coat a 10-inch Bundt pan with nonstick cooking spray and dust it with cocoa powder.
Spread the hazelnuts on a rimmed baking sheet and toast for about 10 minutes, until golden and fragrant. Transfer the hazelnuts to a kitchen towel or paper towel and let cool slightly. Then, rub off the skins using the paper towel and let cool completely. In a food processor, pulse the hazelnuts until they are very finely chopped and slightly powdery. Transfer the ground hazelnuts to a bowl and whisk in the flour and the salt. Set aside.
Sift the 2 cups of confectioners’ sugar with the cocoa powder. In a stand mixer fitted with the paddle attachment, beat the butter with the granulated sugar and brown sugar at medium speed until fluffy, for about 2 minutes.
Scrape down the sides of the bowl and reduce the speed to low. Beat in the whole eggs and egg yolks, one at a time, until just incorporated. Add the vegetable oil and vanilla extract and beat until the batter is uniform in color, for about 1 minute. Add the cocoa mixture and beat until fully incorporated. Using a spatula, gently fold in the hazelnut mixture until no streaks remain.
Scrape the batter into the bundt pan and smooth the surface. Bake the cake in the center of the oven for 40 to 45 minutes, until it starts to pull away from the side of the bundt pan and the top just springs back when you touch it. The center will still be soft. Let the cake cool on a wire rack for at least 2 hours; it may crack and sink slightly but that's ok. Invert the cake onto a plate and dust with confectioners’ sugar and serve. Drizzle some melted chocolate hazelnut butter on top if desired.