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Heat the oven to 200C/180C fan/gas 6. Butter and line a 23 x 32cm Swiss roll tin with baking parchment. Beat the eggs and golden caster sugar together with an electric whisk for about 8 mins until thick and creamy.
Mix the flour, cocoa powder and baking powder together, then sift onto the egg mixture. Fold in very carefully, then pour into the tin. Tip the tin from side to side to spread the mixture into the corners. Bake for 10 mins.
Lay a sheet of baking parchment on a work surface. When the cake is ready, tip it onto the parchment, peel off the lining paper, then roll the cake up from its longest edge with the paper inside. Leave to cool.
To make the icing, melt the butter and dark chocolate together in a bowl over a pan of hot water. Take from the heat and stir in the golden syrup and 5 tbsp double cream. Beat in the icing sugar until smooth.
Whisk the remaining double cream until it holds its shape. Unravel the cake, spread the cream over the top, scatter over the crushed extra strong mints, if using, then carefully roll up again into a log.
Cut a thick diagonal slice from one end of the log. Lift the log on to a plate, then arrange the slice on the side with the diagonal cut against the cake to make a branch. Spread the icing over the log and branch (don’t cover the ends), then use a fork to mark the icing to give the effect of tree bark. Scatter with unsifted icing sugar to resemble snow, and decorate with holly.