Prep Garnishes (Day Before): Make sugared cranberries and rosemary (if using) and meringue mushrooms (if using), allowing for several hours of drying and setting time.
Make Cake: Whisk dry ingredients. Beat egg whites with half the sugar until stiff peaks form. Beat yolks with remaining sugar, oil, and vanilla. Fold egg whites and flour mixture into the yolk mixture gently.
Bake and Roll: Spread batter on a prepared 12x17-inch parchment-lined pan. Bake for 15-16 minutes. Immediately invert the hot cake onto a cocoa-dusted towel/parchment. Peel off the bottom parchment. Tightly roll the cake and towel together from the narrow end. Cool completely (approx. 3 hours).
Make Filling: Whip cold heavy cream, liqueur, sugar, and cocoa until medium-stiff peaks form.
Assemble Log: Gently unroll the cake. Spread whipped cream filling evenly. Sprinkle hazelnuts. Gently re-roll the cake without the towel. Chill for at least 30 minutes.
Ganache and Finish: Make ganache by pouring hot cream over chopped chocolate; stir until smooth. Chill until spreadable (30-60 minutes). Slice off an angled "branch" piece from the log. Coat the log and branch with ganache. Use a fork to create bark texture. Decorate with mushrooms, cranberries, and powdered sugar just before serving.