Preheat oven to 350F. Grease a half sheet pan (18”x13”) and line with parchment paper.
Bring eggs to room temperature. Sift flour, cocoa powder, baking powder, baking soda and salt in a bowl.
In a separate bowl, whip eggs until thick and pale in color. Gradually add sugar while whisking or using an electric mixer. Continue beating until very thick and light.
Add dry ingredients and begin to gently fold into whipped egg mixture. Add oil and vanilla and fold gently until all ingredients are combined. Pour onto prepared sheet pan and smooth with a spatula. Place into preheated oven.
Bake until just set in the center, about 25-30 minutes.
Meanwhile, as the cake is baking, place another sheet of parchment the same size as the cake on your work surface. Dust with powdered sugar.
As soon as you remove cake from the oven, use a knife to scrape down the sides of the pan to easily release the cake. Invert onto sugar-dusted parchment. Immediately roll up cake as pictured above and allow to cool completely.
Once cake is cooled, unroll and cover with frosting to the thickness as pictured above. Roll cake back up. Ends can be trimmed and arranged to look like branches attached to log. Place on serving tray or plate.
Frost entire cake with chocolate frosting, then use a knife or spatula to create wood grain pattern all over cake. Decorate as desired with with meringue mushrooms and powdered sugar or simply rinse fresh rosemary sprigs and cranberries, then dip in sugar, as pictured.