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Heat oven to 190C/170C fan/gas 5. Grease and line a 20 x 30cm Swiss roll tin with baking parchment, then grease the parchment a little too. Put the treacle, syrup, butter and stem ginger in a pan, heat until melted and stir to combine, then set aside to cool a little.
Put the eggs and sugar in a bowl and whisk using an electric hand whisk until light, mousse-like and doubled in size – this will take about 10 mins. The mixture is ready when it holds a ribbon trail from the beaters for 3 secs. Sift over the flour, baking powder and spices, then pour the melted butter mixture around the sides of the bowl so that it trickles down into the whisked eggs. Very gently fold everything together with a large metal spoon. When just combined, pour the mixture into the Swiss roll tin and ease it into the corners. Bake for 12 mins until just cooked.
While the sponge is cooking, lay a sheet of baking parchment, big enough to fit the cake, on your work surface and dust with a little sugar. Once cooked, tip the cake directly onto the parchment. Use a small serrated knife to score a line about 2cm from one of the shorter ends, making sure you don’t cut all the way through – this will help to get a tight roll. Gently roll up from this end, rolling the parchment between the layers. Leave to cool like this on a wire rack to help set the shape.
To make the icing, put the ingredients in a bowl and whisk until smooth. Transfer to a piping bag fitted with a large round nozzle, or use a plastic sandwich bag and snip off one corner to make a hole about 1cm wide. Unroll the sponge and drizzle the surface with 2 tbsp ginger syrup. Pipe a layer of ginger buttercream over the inside of the roll, then use the paper underneath to help tightly re-roll into a roulade. Slice off both ends for a neat finish. The Bûche can be frozen at this point – simply re-roll in the parchment, then in foil, and freeze. Defrost at room temperature before continuing.
Place the Bûche on a serving plate or board. Use the remaining icing to pipe a thick layer over the top of the sponge, zigzagging backwards and forwards to create a tight concertina pattern. Decorate with white pearl sprinkles, if you like. The Bûche will keep in a sealed container for up to 5 days, or can be frozen for up to two months.