Prepare a 10x15 inch jellyroll pan, parchment paper, and a clean flour-sack towel.

Ensure the jellyroll pan is flat to prevent burning.

Cream butter until light and fluffy for the filling.

Beat egg yolks until pale in a separate bowl.

Combine sugar and water in a saucepan and cook to 230 degrees F.

Slowly pour the hot sugar syrup into the yolks while beating until thick and cool.

Beat the sugared yolks into the creamed butter until smooth, then add rum.

Set aside 1/4 cup of the buttercream mixture.

Blend instant coffee with hot water and stir into the reserved buttercream.

Add powdered sugar to the coffee buttercream until spreadable.

Melt chocolate and blend it into the large bowl of buttercream.

Refrigerate both chocolate and coffee buttercreams to chill.

Preheat oven to 450 degrees F.

Line the jelly roll pan with buttered parchment paper.

Sift flour, cocoa, and cornstarch together.

Melt butter.

Beat eggs briefly, then add sugar and beat for 5 minutes until thick and pale yellow.

Fold in the flour mixture, then the melted butter.

Pour into the prepared pan and spread evenly.

Bake for 5-8 minutes until the top springs back when touched.

Sprinkle powdered sugar on the flour-sack towel.

Run a knife around the edges of the cake and invert it onto the towel.

Remove the pan and parchment paper.

Roll the cake up in the towel and let it cool seam side down.

Unroll the cooled cake, remove the towel, and spread half of the chocolate buttercream over the inside surface.

Roll the cake up tightly without the towel, wrap in foil, and refrigerate for at least an hour.

Place the cake on parchment paper on a serving platter.

Slice off both ends of the cake and reserve.

Frost the cake with most of the remaining chocolate buttercream.

Do not frost the ends.

Use a fork to create bark ridges.

Attach the reserved cake ends to the sides to resemble branches and frost them.

Frost the ends of the log and branches with the coffee buttercream.

Optionally, make meringue mushrooms to decorate.