Preheat the oven to 350°F. Grease a 10x15-inch jelly roll pan and line with parchment.
Whisk together the flour, cocoa powder, baking powder, and salt in a bowl.
Beat the eggs and ½ cup sugar together on high speed until pale and tripled in volume, about 4 minutes. Beat in the oil and ½ teaspoon vanilla.
Fold the dry ingredients into the egg mixture in three additions until just combined.
Spread the batter evenly in the pan and bake for 10–12 minutes, until the top springs back when touched.
Immediately invert the warm cake onto a clean kitchen towel dusted with the remaining ¼ cup sugar. Peel off the parchment and roll the cake up in the towel from the short end. Cool completely, about 30 minutes.
Whip the cold heavy cream, 2 tablespoons powdered sugar, and remaining ½ teaspoon vanilla to stiff peaks. Unroll the cooled cake, spread with the whipped cream, and re-roll without the towel.
Beat the softened butter, 3 cups powdered sugar, and melted chocolate together until fluffy to make the frosting. Frost the log and drag a fork along the length to create bark texture.