Heat oven to 350 degrees. Wrap coconut in a kitchen towel, and place in oven for 20 minutes. Remove from oven, and place on a work surface. Reduce oven to 250 degrees. Tap coconut with a hammer until it cracks in several places; break it into pieces. The white flesh should be slightly pulling away from the hard shell. Separate it, keeping pieces of coconut as large as possible. Using a vegetable peeler, shave the largest pieces of coconut into long curls, and place on a baking sheet in oven. Bake for 15 to 20 minutes, until just dry. Grate the smaller pieces on the small holes of a box grater. You will need about 1/2 cup grated coconut for the cake filling.
Heat oven to 400 degrees. Line a 12-by-17-inch jellyroll pan with parchment paper, and set aside. In the bowl of an electric mixer fitted with the whisk attachment, beat egg yolks on high speed until pale yellow, 4 to 5 minutes. Transfer to a medium bowl, and set aside. Wash and dry mixer bowl.
Place egg whites in bowl of electric mixer fitted with the whisk attachment. Beat on medium speed until soft peaks form, 1 to 2 minutes. Increase speed to medium high, and add sugar gradually, whipping until stiff peaks form. Transfer egg-white mixture to a large bowl.
Using a rubber spatula, fold egg-yolk mixture into egg-white mixture. Sift cocoa powder and flour over top, and gently fold in. Pour batter into prepared pan, and smooth top with an offset spatula.
Bake until cake springs back when touched, 9 to 10 minutes. Remove pan from oven, and immediately turn cake out onto a wire rack lined with parchment paper. Peel parchment paper from top of cake, and cool completely.
Place a clean kitchen towel on work surface. Transfer cake on parchment paper to towel. Using a pastry brush, coat cake with rum. Using an offset spatula, spread white-chocolate mousse over cake. Sprinkle with 1/2 cup freshly grated coconut.
With a short end of cake facing you, roll carefully into a roulade. Wrap in a kitchen towel, and secure with clothespins or large metal binder clips to hold the roll in a tight round shape. Place on a baking sheet, and refrigerate until firm, at least 2 hours.
Remove from refrigerator, and unwrap. Using a serrated knife, trim the ends of the roll, cutting on the bias. Choose the more presentable end, and place it, cut-side up, on the log, forming a branch. If desired, use the other cut end for a second branch.
Transfer the log to a serving platter. Slide 5-inch strips of parchment paper under the log to protect the tray while decorating. Coat with seven minute frosting using an offset spatula. Remove parchment pieces. Decorate with meringue mushrooms, snowy chocolate truffles, coconut shavings, and rosemary sprigs.
To make sugared rosemary sprigs, brush washed sprigs with a glaze of one egg white from a large egg thinned with a tablespoon of water, then coat in superfine sugar, and let dry.