Eggnog Semifreddo Genoise Cake with Meringue Frosting

Cake: Preheat oven to 425 degrees. Spray two 12 1/2-by-17 1/2-inch rimmed baking sheets with cooking spray. Line with parchment; spray parchment, then dust with cocoa, tapping out excess. Dust 2 kitchen towels with cocoa.

Whisk together cocoa, flour, and salt. Combine eggs, yolks, and sugar in the heatproof bowl of an electric mixer, then set over (not in) a pot of simmering water. Whisk until sugar is dissolved and mixture is warm to the touch, about 2 minutes. Transfer bowl to mixer fitted with the whisk attachment; beat on medium-high speed 2 minutes. Increase speed to high and beat until mixture is pale and thick, about 3 minutes more.

In two additions, sift cocoa mixture over top of egg mixture, then fold in. When almost incorporated, pour melted butter down side of bowl; gently fold to combine, scraping bottom of bowl to make sure dry ingredients are fully incorporated. Divide between prepared sheets, using an offset spatula to spread evenly and smooth tops. Bake until center springs back when lightly touched, 5 to 7 minutes. One sheet will bake quicker than other, depending on which is closer to heat source; remove that sheet and run a small sharp knife around edges of cake. Invert onto a prepared towel; remove parchment. Immediately, starting at one short side, roll into a log, incorporating towel. Repeat with second cake. Let cool while preparing filling.

Filling: Sprinkle gelatin over rum; let stand until softened, 5 minutes. Combine egg yolks and 3/4 cup sugar in a heatproof bowl set over (not in) a pot of simmering water. Whisk until pale and fluffy, about 2 minutes. Remove from heat; whisk in gelatin mixture. Set bowl over pot of simmering water again and whisk until gelatin has dissolved. Transfer mixture to another bowl.

Whisk cream with nutmeg and vanilla to medium peaks. Fold one-third of whipped cream into gelatin mixture, then fold gelatin mixture into remaining cream. In a clean mixing bowl, whisk egg whites and remaining 1/2 cup sugar on medium-high speed to stiff peaks, about 6 minutes. Fold into cream mixture.

Unroll cooled cakes, remove towels, and transfer each to a separate rimmed baking sheet. Set aside 1 cup filling; immediately divide remaining filling among cakes, spreading it evenly over each. Freeze cakes until filling is starting to firm up but is still pliable, 20 minutes.

Slice each cake in half length-wise to create 4 cake strips. Starting at a short end of one strip, roll up cake and place vertically in the center of a cake stand. Starting where rolled strip ends, wrap next strip around cake. Continue to wrap with remaining strips. Use reserved filling to fill in top of cake. For added stability, wrap a parchment collar around base of cake, keeping bottom of paper flush with cake stand. Freeze until firm, at least 8 hours and up to 2 days. Once frozen, trim top of cake with a serrated knife to level. Keep frozen until ready to frost.

Frosting: Combine egg whites, sugar, and cream of tartar in mixer bowl set over (not in) a pot of simmering water. Whisk constantly until sugar has dissolved and whites are warm to the touch, about 3 minutes (test by rubbing between your fingers).

Transfer bowl to mixer fitted with whisk attachment and beat, starting on low speed and gradually increasing to high, to stiff, glossy peaks, about 10 minutes. Add vanilla and beat until combined.

Spread frosting evenly over top and sides of cake; apply with offset spatula, using an S motion to create a textured appearance. Freeze cake at least 30 minutes and up to 1 day. Use a kitchen torch to brown meringue; serve immediately.