Preheat the oven to 350 degrees F (175 degrees C). Lightly spray a 10x15-inch jellyroll pan and line it with parchment paper. Sift flour with baking powder and salt and set aside.
In a large mixing bowl, beat the eggs on high for several minutes until they are very pale and fluffy. Gradually add the sugar, beating for 1 to 2 minutes more or until very thick. Gently, but thoroughly, fold in the flour mixture.
Melt the chocolate in a small saucepan over low heat. In a small bowl, combine the 2 tbsp of water with the 2 tbsp coffee liqueur, the remaining 2 tbsp sugar, and the baking soda, then gradually stir into the melted chocolate until smooth. Quickly but thoroughly fold the chocolate mixture into the batter.
Pour batter into prepared 10x15 inch pan. Bake at 350 degrees F (175 degrees C) for 18 to 20 minutes, or until a toothpick inserted into the cake comes out clean. Lightly sift an even layer of confectioner sugar over a cloth napkin or tea towel. Flip the cake out of its pan onto the prepared cloth as soon as it comes from the oven. Carefully peel away the parchment paper. Lightly dust the top of the cake with confectioner sugar, then trim away crisp edges. Starting with one of the short sides of the cake, immediately roll the cake up in the cloth, jellyroll style, and cool thoroughly on a rack.
For the Filling and Frosting: In a small saucepan over low heat, melt the chocolate. Remove from heat and let cool to lukewarm. In a medium bowl, beat the cream cheese with the confectioner's sugar until smooth, then blend in the vanilla extract and coffee liqueur. Blend in the melted chocolate. Unroll the cake and spread about 1/3 of the filling evenly over the surface. Roll the cake back up.
Arrange the cake roll on the serving tray, then frost generously, swiping with an icing spatula to form the long 'bark line' design. Swipe the ends of the cake in a circular motion to simulate the tree rings of a cut log. Decorate the log as desired. Yields one cake.