For the genoise, preheat the oven to 180C/160C Fan/Gas 4.
Whisk the egg whites in a bowl (or free-standing mixer) until soft peaks form when the whisk is removed. Gradually add the sugar as you whisk the egg whites, until stiff peaks form when the whisk is removed.
Reduce the speed of the electric whisk to medium, then whisk the egg yolks into the mixture, one by one.
Sift half of the flour into the bowl, then gently fold it into the egg mixture using a spatula. Add the remaining flour and carefully fold it into the egg whites too.
Pour over the melted butter and gently fold it into the egg white mixture.
Line a 30cm/12in Swiss roll tin with silicone paper, sticking each corner to the tin with a dab of the genoise mixture.
Pour the genoise into the prepared tin, pushing it into the corners and levelling the surface using a palette knife.
Bake the genoise in the oven for 10 minutes, or until it is pale golden-brown in colour and springy to the touch. Remove from the oven and set aside to cool in the tin.
Meanwhile, for the buttercream, heat the sugar and 50ml/2fl oz water in a small saucepan until the temperature of this sugar syrup reaches 121C (use a sugar thermometer for this).
Using an electric whisk or free-standing mixer, whisk the egg yolks with 30ml/1fl oz water for 5 minutes on full speed until pale and fluffy.
Reduce the speed of the electric beaters, then drizzle the sugar syrup in a thin stream down the side of the bowl. (Caution: do not pour the hot syrup directly onto the beaters). Whisk until the mixture (called a sabayon) thickens and looks silky, then continue to whisk for a further 3 minutes to allow the sabayon to cool slightly.
Reduce the speed of the beaters to a slow speed, then whisk in the rum. Add the softened butter a little at a time, whisking until each addition has become fully incorporated into the sabayon, then continue to whisk for a further 3 minutes.
Fold in the chopped chestnuts and set aside until needed (chill in the fridge if making in advance).
For the soaking liquid, bring the sugar and 40ml/1¼fl oz water to the boil in a small saucepan, then remove the pan from the heat and stir in the rum.
Using a pastry brush, brush the soaking liquid evenly over the cooled sponge.
Turn the soaked sponge upside down onto a clean tea towel, then carefully peel away the silicone paper.
Spread one-third of the buttercream in an even layer all over the soaked sponge, then roll the sponge as tightly as possible, starting at the long edge that’s closest to you. (To make this easier, fold the starting edge of the sponge over itself, then guide the rest of the Swiss roll tightly around it.)
Wrap the sponge roll tightly in cling film, then chill in the fridge for 15 minutes.
Unwrap the sponge roll and transfer it to a cake board with the seam on its underside. Carefully trim each end using a sharp knife.
Spread the remaining buttercream all over the sponge roll in an even layer using a palette knife (or piping bag, if desired). Drag a fork lightly and unevenly through the buttercream to make it look like tree bark. Refrigerate for a further 30 minutes.
Meanwhile, for the chocolate sauce, bring the milk and cream just to the boil in a saucepan. Remove the pan from the heat and stir in the chocolate until smooth and melted.
Just before serving, dust the bûche de Noël with a little cocoa powder and garnish with grated chocolate or chocolate curls. If desired, slice the end off the log and place it on top. Decorate with whole candied chestnuts.
To serve, slice the bûche de Noël and drizzle with a little of the chocolate sauce.