Preheat the oven to 180C/350F/Gas 4.
Line a 23x33cm/9x13in Swiss roll tin with a sheet of greaseproof paper and brush the paper lightly with vegetable oil.
Whisk the caster sugar and egg yolks together in a bowl until pale and thick. Stir in the flour until combined, then stir in the melted chocolate.
Meanwhile, whisk the egg whites until stiff peaks form when the whisk is removed.
Gently fold a spoonful of the whisked egg whites into the chocolate mixture, using a large metal spoon, then gently fold in the remaining egg whites until just combined. (NB: Do not overmix.)
Carefully pour the cake batter into the prepared Swiss roll tin and bake for 18-20 minutes, until the cake has risen and springs back when a finger is pressed gently into the centre.
When the cake is cooked, set it aside, in the tin, on a wire rack, to cool for at least two hours.
When the cake is completely cool, run a butter knife around the inside edge of the tin to loosen the cake. Spread a tea towel out on a large chopping board. With one bold movement, turn the whole cake out onto the tea towel, then lift away the tin. Carefully peel away the greaseproof paper lining.
Meanwhile, for the filling, carefully mix the chestnut purée, whipped cream and mascarpone in a bowl until well combined, then fold in the chopped chestnuts. Spread the filling all over the cake, leaving a 2cm/1in border free at the edge.
Starting at the longest side of the cake, gently but firmly roll up the cake, as you would a Swiss roll, using the tea towel as a guide. Transfer the Swiss roll to a flat serving dish.
To serve, cut the yule log into slices and dust with icing sugar.
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James is joined by chefs Paul Rankin and Nigel Haworth, plus actor Bradley Walsh.
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