Cake: Preheat oven to 350 degrees. Butter two 10-inch springform pans (or one 10-inch springform pan and one 10-inch round cake pan), and line with parchment. Butter paper; flour paper and pans. Whisk together 1/3 cup sugar, the flour, cocoa powder, baking powder, baking soda, and salt in a bowl. Mix oil, water, and vanilla in another bowl.
Beat egg yolks with a mixer on medium-high speed until pale and thick, about 5 minutes. Gradually add oil mixture, beating until combined. Gradually beat in flour mixture until combined.
Beat egg whites on medium-high speed in another bowl with clean beaters until foamy, gradually adding remaining 1/3 cup sugar, until medium-stiff peaks form. Fold a third of the whites into yolk mixture with a rubber spatula, then gently fold in remaining whites.
Divide batter between pans, and spread evenly with an offset spatula. Bake until cakes are set and spring back when touched, 10 to 12 minutes. Let cool completely in pans on a wire rack. Remove cake from 1 pan (the cake pan, if using), and crumble.
Marshmallow swirl: Place egg whites, sugar, and cream of tartar in a bowl set over a saucepan of simmering water. Whisk constantly until sugar dissolves and mixture is warm, 3 to 4 minutes. Remove from heat. Beat with a mixer, starting on low speed and gradually raising to high, until stiff, glossy peaks form, 5 to 7 minutes. Beat in vanilla until thoroughly combined. Use immediately.
Assembly: Meanwhile, stir together ice cream and hazelnuts in a large bowl. Fold in three-fourths of the marshmallow swirl and all the crumbled cake. Just barely fold in remaining marshmallow swirl, and spread over cake in pan, swirling top. Freeze until firm, at least 6 hours and up to overnight. Remove side of pan, and let sit at room temperature 5 minutes. Cut into wedges, and serve.