Preheat the oven to 350°F (175°C). Line a baking sheet that is around 10×15 inches with parchment paper and set aside.
In a mixer fitted with the whisk attachment, whisk the egg whites until foamy.
Add ½ the amount of sugar and continue to whisk until stiff peaks form. Transfer to another bowl and set aside.
Switch the mixer attachment to the paddle. Using the same mixing bowl (no need to wash it), beat the egg yolks and remaining sugar until light and silky. About 5 minutes.
Beat in the oil and baking powder until well incorporated. Then beat in the flour.
Gently fold the egg whites into the yolk mixture.
Pour the batter into the prepared baking sheet and spread the batter evenly.
Bake until lightly golden brown and a toothpick comes out clean. About 15-20 minutes.
Meanwhile, lay out a clean dish towel on a working surface and sprinkle with granulated sugar.
Once the cake is out of the oven, carefully flip it onto the sugared dish towel and remove the parchment paper.
Use the dish towel to quickly roll the cake while it is still warm, starting from the short edge. The towel will roll into the cake. Let cool completely.
Remove the packaging from the ice cream so that you have the block of ice cream still in tact. Cut the block into slices that are about 1 inch thick (you may not need to use the entire block).
Once cooled, carefully unroll the cake. Line the entire cake with the slices of ice cream.
Tightly roll the cake back up, without the towel. Wrap the log tightly with plastic wrap and place in the freezer until completely solid.
Melt the chocolate and butter using a microwave or a double boiler. Pour the chocolate over the cake and spread it evenly. Place back in the freezer to harden. Run a fork lengthwise along the cake to make textured lines that resemble tree bark.
Consider decorating with these meringue mushrooms!