Grease a 9×15 inch sheet pan (0.5 inch in height), and line the bottom and the sides with parchment paper. Preheat the oven to 190 degrees Celsius.
Sift the flour and cocoa powder together thrice. Set aside.
Place the eggs in a metal pot. Whisk in the sugar until combined. Set the pot over gentle heat and whisk constantly, until the mixture is foamy and the temperature is warm, that of baby’s milk. Remove from heat and transfer the mixture into a mixing bowl.
Whip the mixture with an electric mixer on high speed until the mixture is no longer foaming and the bubbles disappeared. Whip until the mixture has tripled in volume and is thick and rich. Switch to the lowest speed and whip for 1 minute, until the mixture leaves a trail for a short while, before merging with the mixture.
Sift the flour mixture over the egg mixture in three parts, gently folding the flour into the egg mixture until thoroughly combined. There shouldn’t be any flour pockets visible and do not over-mix.
Pour a couple tablespoonfuls of cake batter into the hot, melted butter and fold until the butter has been incorporated into the cake batter. Pour the mixture back into the cake batter and fold it in gently.
Transfer the cake batter into the sheet pan. Tilt the sheet pan so the batter spreads out evenly and drop it several times on the counter top to burst the uneven air bubbles.
Bake in the middle rack for 10 to 15 minutes until the cake springs back when touched gently and a skewer inserted into the centre of the cake comes out dry.
While the cake is baking, sprinkle a clean tea towel with granulated / caster sugar. When the cake is done, remove it from the oven and set it aside for 1 minute. Turn the sponge out onto the tea towel with sugar. Roll up the cake firmly from the shorter end and set it aside to cool completely.
Peanut Butter Frosting
In a medium bowl, use an electric mixer and beat the icing sugar, butter and peanut butter until smooth and creamy. Set it aside.
Chocolate Meringue Buttercream
Place the dark chocolate in a heatproof bowl. Place the heatproof bowl over a pan of simmering water, ensuring that the water do not touch the bottom of the heatproof bowl. Stir the chocolate occasionally until all the chocolate has melted. Alternatively, microwave the chocolate on medium-high, checking the chocolate every 30 seconds, until the chocolate looks shiny. Stir the chocolate until all has melted, heating it if needed. Leave the bowl of chocolate aside to cool to room temperature, stirring occasionally.
Place the egg whites in a different heatproof bowl. Place the bowl over a pan of simmering water, again ensuring that the water do not touch the bottom of the bowl. Use a whisk and whisk the egg whites, adding the sugar in 3 batches. Whisk the egg whites + sugar mixture until the mixture is warm like baby's milk and that all the sugar has dissolved. Test it by dipping your fingers into the egg whites and rubbing it against your fingers to ensure all sugar has dissolved.
Remove the egg whites mixture from the heat. Using an electric beater now, beat the egg whites mixture on high speed until the egg whites had cooled down completely and the mixture is thick and glossy (stiff peaks). The egg whites will not slide even when the bowl is overturned.
Beat in the unsalted butter cube by cube . The mixture will become very watery and seems like the buttercream has failed. But fret not! After adding all of the butter, the mixture will gradually come together and become a creamy and smooth buttercream. Add in the cooled, melted chocolate (hot, melted chocolate may end up cooking the eggs, so make sure it's completely cooled!). Beat with the electric mixer on low speed until fully incorporated.
Assembly
Unroll the cooled swiss roll. Spread an even layer of Peanut Butter Buttercream on the swiss roll and roll it up to form a log. Cut out a short section of the swiss roll at a diagonal to from the branch.
Spread some Chocolate Meringue Buttercream on one end of the branch and join it to the remaining section of the swiss roll. Spread a thin layer of Chocolate Meringue Buttercream all over the log cake. Refrigerate it for 10 minutes.
Spread an even layer of the remaining Chocolate Meringue Buttercream all over the log cake. Refrigerate for 30 minutes.
Using a fork, draw lines all over the log cake and use a skewer to draw a spiral on the ends of the log cake. Decorate with Christmas decorations. Refrigerate until ready to serve.