Place chocolate in a small bowl with 1/4 cup water. Set bowl in a pan of barely simmering water, stirring frequently until melted and smooth. Set aside to cool. Combine 1/2 cup, or more to taste, of chestnut cream into 1 slightly rounded cup of buttercream. Set aside.

Assemble the Buche: Peel parchment liner from cake. Lift foil under cake and invert it onto a second sheet of foil. Remove the original foil, peeling away the browned genoise "skin" stuck to it. Moisten the cake with the rum, using a pastry brush. Spread chestnut buttercream evenly over the cake. Roll the cake tightly, the long way, into a "log" using the foil to aid you if necessary. Wrap log tightly in foil. Freeze until hard, or at least firm. The firmer the cake, the easier it will be to decorate. Log may be completed to this point, and kept frozen, up to 6 weeks. Stir chocolate mixture into remaining buttercream and refrigerate or freeze until needed. Finish and Decorate the Buche: Soften reserved chocolate buttercream and beat just until smooth. Unwrap the frozen log. Trim a fraction of an inch from each end to even the edges. Spread the log with most of the reserved chocolate buttercream. Do not frost the ends. Use a fork to texture the buttercream so that it resembles the bark of a tree. Dip a sharp knife in hot water, then wipe dry. Cut the diagonal "stump" off each end of the log. Set stumps aside. Transfer log to a serving platter. Affix stumps to the log. Cover bare spots with buttercream. Use the fork to touch up the bark effect as necessary. Decorate with sprigs of pine and meringue mushrooms. Store in the refrigerator. For best flavor and texture, remove Buche from the refrigerator at least an hour before serving.