Preheat oven to 375°. Prepare towel for cake rolling step by sifting a generous amount of powdered sugar on top of clean kitchen towel.
Grease a 15 1/2 X 10 1/2 X 1-inch pan, then line with waxed paper or aluminum foil and grease again. Combine flour, baking powder and salt, then set aside.
On high speed, beat eggs in small bowl until very thick, about 5 minutes. Pour eggs into larger bowl and beat in 1 cup sugar, 1 tablespoon at a time. Reduce speed to low and beat in vanilla and coffee-flavored liqueur or water. Add flour mixture, 1 tablespoon at a time, mixing just until batter is smooth and all flour is incorporated. Pour into prepared pan, making sure the batter is spread to corners. Bake 12 to 15 minutes, until light brown.
Loosen cake from edges and invert onto prepared towel. Remove waxed paper or aluminum foil. While still hot, roll cake and towel from narrow end. Cool on wire rack for 1 hour.
When cake has cooled, prepare filling. Beat heavy cream, 1 tablespoon sugar and instant espresso in chilled bowl until stiff peaks form. Unroll cake and remove towel. Spread whipped cream over cake and re-roll. Place cake on serving platter. To form log, cut off a 3-inch diagonal slice from one end and attach to one side of cake using frosting. Frost the rest of the cake. To decorate and create bark effect, use tines of fork and gently slide along top and sides of cake so it resembles a tree branch.
Sift cocoa, then beat beat with butter on medium speed until creamy. Sift powdered sugar, then add to cocoa mixture, beating on low speed. Beat in vanilla and 1 tablespoon hot water, mixing until smooth. Add up to 1 tablespoon more hot water until frosting is of spreading consistency.