1) Prep the pan
Heat oven to 350°F. Line a 10x15-inch rimmed sheet pan with parchment, then lightly grease the parchment (especially the corners). Set a clean thin kitchen towel on the counter and dust it generously with powdered sugar or cocoa powder.
2) Whip the eggs and sugar
In a stand mixer (or large bowl with a hand mixer), beat eggs, granulated sugar, and vanilla on high for 6 to 8 minutes, until very pale, thick, and the batter ribbons off the whisk.
3) Fold in dry ingredients
Sift cocoa powder, flour, baking powder, and salt over the egg mixture. Fold gently with a spatula until no dry streaks remain. Drizzle in the melted butter and fold just until combined. Be gentle. You are protecting the air you whipped in.
4) Bake
Spread batter evenly in the pan. Bake 10 to 12 minutes, until the top springs back when lightly pressed and the cake looks set. Do not overbake.
5) Roll while warm
Immediately loosen edges with a knife. Carefully lift the hot pan (oven mitts, please) and invert the cake onto the prepared towel. Peel off the parchment slowly, then dust the surface lightly with more powdered sugar or cocoa. Starting from a short end, roll the cake up snugly in the towel. Let it cool completely, seam-side down.
6) Make the mascarpone filling
Whip cold cream and powdered sugar to soft peaks. In another bowl, beat mascarpone with vanilla and a pinch of salt just until smooth. Fold the whipped cream into the mascarpone in two additions until fluffy.
7) Fill and re-roll
Unroll the cooled cake carefully. Spread filling in an even layer, leaving about 1/2 inch bare on the far edge (it will squish forward as you roll). Roll back up gently but firmly, without the towel.
8) Create the log shape
For the classic look, slice a 2 to 3-inch piece off one end on a diagonal. Place the main roll seam-side down on your serving platter. Attach the small piece to the side like a branch, using a little filling as “glue.” Chill 20 minutes to set.
9) Make ganache
Place chopped chocolate in a bowl. Heat cream in a small saucepan until steaming and just barely simmering at the edges. Pour over chocolate, add corn syrup or honey (if using) and salt, then let sit 2 minutes. Stir from the center outward until smooth.
Troubleshooting: If it looks split, let it sit 1 minute, then whisk gently until it comes together. If it feels too thin, cool a bit longer. If it gets too thick to spread, warm it for a few seconds over a bowl of warm water, stirring often.
Cool 10 to 20 minutes until thick and spreadable.
10) Frost and make bark
Spread ganache over the cake (including the branch) with an offset spatula. Drag a fork or the tip of the spatula through the ganache to create bark lines. Chill 30 minutes to set.
11) Decorate and serve
Right before serving, dust lightly with powdered sugar. Add rosemary sprigs and cranberries if you like. Slice with a serrated knife, wiping the blade between cuts for clean spirals.
