Grease a 33 x 23 cm (13 x 9 inch) Swiss roll tin and line it with nonstick baking paper. Whisk the eggs and sugar in a heatproof bowl over a pan of hot water until the mixture leaves a trail when the whisk is lifted. Sift in the flour and cocoa powder and fold in.
Pour into the prepared tin and spread into the corners. Bake in a preheated oven, 180°C (350°F), Gas Mark 4, for about 15 minutes until just firm. Invert the cake on to a sheet of nonstick baking paper dusted with icing sugar. Peel away the paper that lined the tin, then roll the sponge in the fresh paper and leave to cool.
Whip half the cream until softly peaking, then fold in the chestnut purée. Unroll the sponge and spread the chestnut cream over the top (don't worry if the cake cracks). Roll the cake back into a log shape.
Bring the remainder of the cream almost to the boil. Remove from the heat and stir in the chocolate pieces. Leave until melted, then stir until smooth. Allow to cool.
Arrange the cake, seam side down, on a serving plate. Lightly whip the chocolate cream, then spread it over the top and sides of the cake, and mark to look like tree bark. Dust with icing sugar just before serving.