Preheat your oven to 350 degrees Fahrenheit. Grease a standard jelly roll pan and line it with parchment paper, ensuring the paper hangs over the edges slightly for easy removal. Dust a clean kitchen towel with cocoa powder and set it aside for later use.
In a large bowl, beat the eggs and granulated sugar on high speed for about 5 to 7 minutes. You want the mixture to become very pale, thick, and tripled in volume. This aeration is what gives the sponge its lightness since there is no chemical leavening like baking powder in this recipe.
Sift the flour, cocoa powder, and salt together in a separate bowl. Gently fold the dry ingredients into the egg mixture using a spatula. Be very careful not to deflate the air you just whipped in. Stir in the alcohol-free vanilla until just combined.
Spread the batter evenly into the prepared pan. Bake for 12 to 15 minutes, or until the top of the cake springs back when lightly touched. Do not overbake, as a dry cake will crack more easily during the rolling process.
While the cake is still hot, immediately turn it out onto the cocoa-dusted towel. Carefully peel off the parchment paper. Starting from the short end, roll the cake and the towel together tightly. Let it cool completely in this rolled position, which trains the cake to hold its shape.
In a chilled bowl, whip the heavy cream and powdered sugar until stiff peaks form. If you want a more stable filling, you can fold in a little bit of softened cream cheese or mascarpone at this stage. Keep the filling refrigerated until the cake is completely cool.
Gently unroll the cooled cake. Spread the whipped cream filling evenly over the surface, leaving a small border at the edges. Carefully reroll the cake, this time without the towel inside. Place the roll seam-side down on a serving platter.
Melt the chopped chocolate and butter together until smooth to create a thick ganache. Let it cool until it reaches a spreadable consistency. Spread the chocolate over the cake roll. Use a fork to drag lines through the frosting to create a bark-like texture. Trim the ends for a clean look, or cut a small piece off at an angle and attach it to the side to look like a branch.
