Preheat the oven to Gas Mark 6, 200°C, fan 180°C. Line a 33 x 23cm (13 x 9in) Swiss roll tin with nonstick baking paper.
Whisk the sugar and eggs with an electric whisk for 10 minutes, until pale and thick. Sift the flour and cocoa into the mixture, plus 1 tbsp of lukewarm water. Gently fold into the mixture, using a large metal spoon, until evenly mixed.
Pour the mixture into the tin and use a spatula to smooth gently into the corners. Bake in the centre of the oven for 10-12 minutes until risen and just firm to the touch.
Meanwhile, lay out a clean, damp tea towel. Place a piece of nonstick baking paper larger than the sponge on top and dust with caster sugar. Turn the sponge onto the paper and peel the paper off the base.
Make an incision line about 1cm (1/2in) from one short edge, being careful not to cut through the cake. This will help you start rolling. Use the paper to roll the sponge tightly. Wrap in the tea towel. Allow to cool, then unroll carefully and spread the cake with whipped cream and roll back up again.
To make the icing, melt the chocolate and butter in the microwave. In a large bowl combine the icing sugar, vanilla and 120ml (4fl oz) of the milk. Add the melted chocolate mixture and stir well.
Add the remaining milk, a little at a time, until you have the desired consistency. Let the icing stand until spreadable, it will thicken as it cools.
Spread the icing over the Yule log and dust with icing sugar.
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