Place the rosemary sprigs in a medium heatproof bowl and set aside. In a medium saucepan set over medium heat, bring the water and 1/2 cup of sugar to a simmer, whisking until the sugar has dissolved. Remove the pan from the heat and allow to cool for a few minutes. Pour the sugar syrup over the rosemary and stir. Cover the bowl and set aside for 15 minutes. Line a baking sheet with parchment paper. Using a slotted spoon, transfer the rosemary from the sugar syrup and place it on the prepared baking sheet. Allow to dry, uncovered, for 1 hour. Pour remaining sugar into plastic container provided with a lid. Add the rosemary sprigs and gently shake the container, coating the sprigs in the sugar. Remove the sugared sprigs from the container, place them on a parchment paper and let them dry, uncovered, for at least 1 hour at room temperature or in the refrigerator. You can then cover them and store in the refrigerator for up to 3 days.NOTE: ideally, you make these the day before
Make the cake:
Preheat oven to 350°F.
Generously butter a parchment-lined 12×17-inch cookie sheet (you can previously butter the cookie sheet itself, so the parchment paper sticks). You need an extremely nonstick surface for the thin cake to roll.
Whisk the flour, cocoa powder, baking powder and salt together in a medium bowl. Set aside.
Using a hand mixer or a stand-up mixer fitted with the whisk attachment, beat the egg whites and 1/2 cup sugar on high speed for 4-5 minutes or until stiff peaks form. Transfer to another bowl.
Using the same mixing bowl (no need to clean it), add the egg yolks, remaining sugar, oil and vanilla extract. Beat on high speed for 3-4 minutes or until thickened and light in color.
Add half of the whipped egg whites into the egg yolk mixture. Beat on low speed for 10 seconds. Repeat with remaining egg whites and beat on low for 10 seconds.
Add half of the flour mixture and beat on low or fold with a silicone spatula until combined. Repeat with remaining flour mixture. Avoid over-mixing so egg whites don’t deflate. Batter will be very light.
Spread batter evenly into prepared pan. Gently tap the pan on the counter a couple times to pop any air bubbles. Bake for 15-16 minutes, checking a couple of minutes before the 15 minute mark to avoid over-baking the cake.
As the cake bakes, place a thin kitchen towel flat on the counter. Place a piece of parchment paper larger than the cake on top of the kitchen towel. Using a fine mesh sieve, dust parchment with some cocoa powder.
Once the cake comes out of the oven, quickly run a knife around the edges to loosen it. Immediately invert it onto the parchment/towel. Peel off the parchment paper that was on the bottom of the cake as it baked.
Roll the cake: begin tightly rolling the hot cake up with the parchment/towel. Do this slowly and gently. Allow the cake to cool completely rolled up in the parchment/towel. You can place it in the refrigerator to speed this process up, about 3 hours and up to 1 day.
Prepare the chocolate mousse filling:
Whip the cream until soft peaks form, and gradually add the cocoa powder, one tablespoon at a time. Add in the icing sugar and the vanilla extract and continue to whip until stiff peaks form.
Remove the cake roll from the refrigerator and allow to sit on the counter for a few minutes to warm.
Gently and very slowly unroll the cake. Spread the chocolate mousse evenly on top, leaving about a 1/2 inch border around the cake. Use a small offset spatula to spread. Gently roll the cake back up, without the parchment/towel this time. Roll it slowly. This part can be a bit messy.
Carefully place the rolled up cake on a cutting board. If the exterior of the cake looks moist, dust with a little cocoa powder. Cover with plastic wrap and refrigerate for at least 30 minutes and up to 2 days before shaping and topping with ganache.
Make the ganache topping:
Place the chopped chocolate in a medium bowl. Heat the cream in a small saucepan until it begins to simmer. Pour over the chocolate and let it sit for 2-3 minutes to gently soften the chocolate. Slowly stir until completely combined and chocolate has melted. Refrigerate, uncovered, for at least 30 minutes and up to 1 hour to thicken.
Remove rolled cake from the refrigerator and gently place on a platter. Diagonally slice a 0.5-1 inch section off each end. Place each section on each side of the roll, forming two branches (you can use 1 Tbs of the chocolate ganache as "glue")
Spread the thickened ganache all over the top and sides of the cake, leaving the cut ends exposed. Use a fork to make textured lines resembling tree bark. Decorate with sugared rosemary sprigs and a dusting of confectioners’ sugar just before serving.
Cover leftover cake and store in the refrigerator for up to 3 days.
Notes
Many of the steps can be made ahead:
Make the sugared rosemary ahead of time.
You can make the roll cake in advance because it must cool/chill completely before filling. Refrigerate for at least 3 hours and up to 1 day.
You can also refrigerate the filled cake roll for up to 2 days.
You can make the chocolate mousse 1 day in advance (cover and refrigerate until ready to use).
You can make the chocolate ganache topping 1 day in advance. Cover and refrigerate. However, if you let it sit for longer than 1 hour, it will thicken too much. To thin it out when ready to use, stir and warm it in a heat-proof bowl over a pan of simmering water for about 1 minute. Do not microwave it.
Entire cake can be made 1-2 days in advance too. Cover and refrigerate until ready to decorate and serve.
You can also freeze the assembled cake before or after adding the ganache. Cover and freeze for up to 3 months. Thaw in the refrigerator before continuing with the recipe. It’s best to decorate with sugared rosemary sprigs and a dusting of confectioners’ sugar closest to serving.