Preheat the oven to 375⁰F/190⁰C. Spray baking pan with borders with cooking spray and line with parchment paper. Spray the paper with cooking spray.
Make the batter. Mix egg yolks and ½ cup of granulated sugar in a bowl of a stand mixer fitted with a whisk attachment on high speed until the foam is 3 times its original volume and falls in thick ribbons when the whisk is lifted. Transfer to a large mixing bowl, set aside.
Place egg whites in a clean bowl and beat them with a clean whisk attachment on medium speed until frothy. Gradually add the remaining ¼ cup of granulated sugar and continue to whip until medium peaks form when the whisk is lifted.
Fold the egg whites into the yolk mixture with a rubber spatula in 2 or 3 additions. Scatter all flour, vanilla, and salt over the batter and fold it just long enough to blend the flour evenly.
Bake. Pour the batter into the prepared pan and spread gently with a spatula. Bake for 12-15 minutes until the top is golden brown and the edges star to pull away from the sides of the pan.
Make the roll. While the cake is baking, sift confectioners’ sugar over a clean flat-weave towel. When the cake is finished baking, insert it onto the sugar coated cloth. Immediately remove the pan and parchment paper and, starting at a long edge, roll the cake into a log with the towel inside it. Let cool completely.
Make the buttercream.
Cut the butter into 1 tbs chunks and set aside.
Combine the granulated sugar, egg whites, and salt in a small pan and cook on the lowest heat setting whisking constantly until the egg whites are hot to the touch (for about 4 minutes). Pour the mixture into a bowl of a stand mixer and whisk on high speed until it becomes very stiff – for about 5 minutes.
Add butter 1 chunk at a time, beating on medium speed waiting for each piece to fully incorporate before adding another one. Add coffee and beat until all incorporated. Set aside.
Make ahead: you can make butter cream a few days before using. Store in an air-tight container in the fridge. Bring to room temperature and beat with mixer till smooth before using.
Make meringue mushrooms.
Preheat oven to 175°F (80°C). Line a baking sheet with parchment paper. Sift together confectioners’ sugar and flour, set aside. In a small bowl combine egg white and cream of tartar and beat with whisk attachment until the egg white holds soft peaks. Slowly add granulated sugar and beat until very stiff. Add flour mixture and mix with a rubber spatula, working very quickly. Place the mixture into a plastic bag, cut a little corner out and pipe out mushroom hats and legs to the prepared baking sheet. Bake for at least 4 hours.
Make coffee syrup.
Combine coffee and sugar until sugar is dissolve. Let cool before using.
Preheat the oven to 350°F (175°C). Toast almonds in the oven for about 5 minutes, stirring a couple times with a wooden spoon. Place chocolate in a small heatproof bowl. Heat up the cream to just before boiling and pour over the chocolate. Wait a couple minutes before mixing everything until all chocolate is dissolved. Mix in the almonds. Refrigerate until the mixture thickens (for at least 15 minutes).
Process pistachios in a coffee grinder until powdered.
Assemble the log.
Place the cake on a plastic wrap. Apply the syrup with a brush. Spread the butter cream over and gently roll into a log. Refrigerate for about 2 hours. Using a sharp knife, cut diagonally edges of the log. Transfer it to the serving plate. Add the edges to the sides of the log. Apply the bark, and place mushrooms on top attaching the hats to the legs with a little bit of the “bark” cream. Sift a little bit of “snow” confectioners’ sugar and chocolate powder on top. Sprinkle moss over the log and around it.