Directions
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Preheat oven, prep baking sheets:
Preheat oven to 425°F. 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.
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Combine dry ingredients:
Whisk together cocoa, flour, and salt.
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Whisk together eggs, yolks, and sugar:
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.
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Beat mixture until pale and thick:
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.
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Add dry ingredients to egg mixture, add butter:
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.
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Divide batter between sheets and bake:
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.
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Remove from sheets and roll each in a towel:
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.
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Make the 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.
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Whip cream with nutmeg and vanilla
Whisk cream with nutmeg and vanilla to medium peaks.
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Fold whipped cream into gelatin mixture:
Fold one-third of whipped cream into gelatin mixture, then fold gelatin mixture into remaining cream.
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Whisk egg whites and fold into cream mixture:
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.
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Spread filling over cooled cakes; freeze:
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.
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Assemble cake:
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.
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Fill in top of cake with more filling; add parchment collar to cake and freeze:
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.
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Combine whites, sugar, and cream of tartar for 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).
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Whisk frosting:
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.
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Spread frosting over cake; freeze; torch before serving:
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.
