Grease the base and sides of a 10 x 15-inch jelly roll pan or rimmed baking sheet, then line the base with parchment paper. Preheat the oven to 350F. Put the eggs and the sugar in a large bowl and whisk with an electric mixer at medium speed until thick, mousse-like, and doubled in volume. It will take 5 minutes or so.
Mix the flour, salt, and cocoa in a bowl, then sift them over the whisked eggs. Fold them in using a large metal spoon or spatula, cutting and lifting rather than stirring. This will preserve the air bubbles trapped inside the batter. It might take longer than you expect to get the batter to an almost even brown.
Holding the bowl just above the prepared baking pan (pouring from a great height will knock the bubbles flat), pour in the cake batter. Tilt the pan slowly from side to side, letting the batter run into the corners.
Bake for 15 minutes, or until the cake has risen all over and the sides are shrinking away from the edge of the pan. It's best to loosen the cake away from the pan carefully with an icing spatula now, in case it has stuck a little.
Spread a large sheet of parchment paper on the work surface, then sift the 2 tablespoons cocoa powder over it. Flip the cake onto this sheet, remove the pan, then cover the cake with a clean dish towel and let cool. The cloth will trap moisture as the cake cools, helping it stay soft and moist.
Make the filling. Chop the chocolate into small pieces. Heat 1-1/4 cups of the cream in a pan until it bubbles at the edges. Take off the heat and stir in the chocolate, Nutella, and vanilla. Let it melt until smooth, then set aside until it has cooled, but is still fairly liquid.
Pour the rest of the cream into a bowl, add about 1/2 cup of the chocolate ganache, the whisk until very thick, but not stiff.
When the cake has cooled, remove the dish towel, then carefully peel off the parchment paper, Trim about 1/2 inch off each edge with a serrated knife. Score a line in the cake about 1 inch in from the nearest long end.
Spread the filling over the cake, the roll it up from the scored end. Use the paper to help make a tight roll.
Roll the cake onto a clean piece of parchment. Cut about 4 inches off one end, at an angle.
Transfer the large piece to a serving plate, then nestle the shorter one against it to make a branch. Spread the remaining ganache all over, adding plenty of texture with the knife. Chill for at least 1 hour or up to 3 days.
Remove from the fridge 30 minutes before eating and dust with confectioners' sugar to serve.