1) Preheat the oven to 350°F. Line a 10" x 15" jelly roll pan with parchment paper.
2) To make the cake: Beat the eggs and oil until thick and pale.
3) Mix in the sugar, salt, and vanilla.
4) Sift together the flour, cocoa, and baking powder, and fold gently into the egg mixture.
5) Spread the batter into the prepared pan.
6) Bake the cake for 10 to 12 minutes, until the top springs back when pressed.
7) Remove the cake from the oven, and cool it in the pan for 2 minutes.
8) Loosen the cake's edges, and transfer it to a towel dusted with confectioners' sugar. Roll the cake up from the short end. Cool completely.
9) To make the filling/frosting: Beat the butter, chocolate, sugar, and oil at low speed until light and fluffy.
10) To assemble the cake: Unroll the cake, spread it with some of the filling, and sprinkle with the crunch. Re-roll the cake and wrap in plastic. Refrigerate until firm.
11) Cut off 1/3 of the roll. Stand the larger piece upright. Position the smaller piece to look like a branch coming out of the log. Spread the remaining filling over the log. Serve immediately, or refrigerate until ready to serve.
Yield: 10 servings.