Prepare the chocolate ganache. In a small saucepan, heat the cream until simmering. Remove from heat. Add the chocolate to a medium heatproof bowl. Pour the hot cream over the chocolate and let sit for 3 minutes. Gently stir with a spatula until the chocolate is completely melted and smooth. Let the ganache stand at room temperature until it becomes firm, with a spreadable texture.
While the ganache is cooling down, prepare the cake. Preheat the oven to 350°F. Line a rimmed half-sheet pan with parchment paper. Grease parchment paper and the sides of the pan.
In a large bowl, whisk together flour, cocoa, baking powder, and salt. Set aside.
In a large bowl, beat together eggs and sugar until thick and pale. About 4 minutes. Add vanilla and mix until incorporated. Sift one third of the flour mixture into the egg mixture. Gently fold. Repeat the process until all the flour mixture is incorporated. In a small bowl, mix buttermilk and oil. Fold into the egg mixture until well incorporated.
Pour the batter into the prepared pan; gently smooth the top with an offset spatula. Bake 8-9 minutes, until the cake springs back when touched. Be careful of not over baking the cake. Cool the cake in the pan for 3 minutes. Run a knife around the edge of the pan. Lay a clean kitchen towel over the counter and sift 1 tablespoon of cocoa powder over it. Invert the cake onto the towel. Remove parchment paper. Roll the cake and the towel together up from the long side. Cool the cake completely, rolled up.
Prepare the espresso filling. In a medium bowl, beat together both cheeses until smooth. Dissolve the espresso powder in water. Add to the cheese mixture and mix until well incorporated. Add sugar and vanilla and mix until smooth. Slowly add cream and beat until smooth.
Unroll the cake and evenly spread the espresso filling over the cake, almost to the edge. Gently re-roll the cake around the filling. Place the cake, seam-side down, on a serving platter. Refrigerate for 30 minutes. Remove from the refrigerator. Cut a piece of cake, about 5-inches, diagonally. Attach to the bigger piece of cake, simulating a tree branch.
Spread the chocolate ganache all around the cake. Using a fork, create lines over the cake, resembling bark. Refrigerate.
Prepare the meringue mushrooms. Preheat oven to 225°F. Line 2 baking sheets with parchment paper.
In a large bowl, beat the egg whites, at medium speed, until foamy. Add cream of tartar, salt, and vanilla. Beat the egg whites until they reach soft peaks. Gradually add the sugar and beat until the meringue holds stiff peaks. Place half of the meringue in a pastry bag fitted with a round tip. Squeeze the meringue firmly into a disk like shape, mounding the meringue to resemble a mushroom cap. Mold the “caps” with a wet fingertip to create an even finish. Fill the pastry bag with the rest of the meringue and pipe the stems of the mushrooms on the second pan. Squeeze out meringue and pull the pastry bag up, perpendicular to the baking sheet. Bake for 1 hour or until dry to the touch. Turn off the oven and leave the mushrooms inside for one more hour. Remove from oven and lightly sift cocoa on top.
To assemble the mushrooms, poke a small hole at the bottom of the cap. Add a little of the melted chocolate inside. Insert the stem to the cap and press lightly. Repeat the process with the rest of the mushrooms.
Remove the yule log from the refrigerator and decorate with mushrooms and powdered sugar. Serve.