Birch de Noel

Preheat oven to 350°F. Test each of the three eyes at stem end of coconut to find softest one. Use an ice pick and a hammer to pierce it, then drain the juice into a glass through a fine sieve and reserve for another use. Tap coconut with hammer until a palm-size piece of shell breaks off. Continue tapping around broken edge until shell begins to break apart. Place broken shell on a baking sheet; bake about 10 min- utes to loosen flesh. Remove from oven and let cool slightly. Pry out flesh with a paring knife. With a potato peeler, remove any remaining brown skin from flesh.

Reduce oven temperature to 250°. Using a vegetable peeler, shave largest pieces of coconut into long curls, place on baking sheet, and bake until just dry but no color has developed, 15 to 20 minutes. Meanwhile, grate smaller coconut pieces on the small holes of a box grater. (You will need about 1/2 cup grated coconut for the filling.)

Increase oven temperature to 400°. Coat a 12 1/2-by-17 1/2-inch rimmed baking sheet with cooking spray. Line with parchment paper; coat with cooking spray. Dust with cocoa; tap out excess. Whisk together cocoa powder, flour, and salt in a bowl. In a heat-proof mixer bowl set over (not in) a pan of simmering water, whisk together eggs, yolks, and granulated sugar until sugar has dissolved and mixture is warm to the touch. Attach bowl to mixer fitted with the whisk attachment and beat on medium-high speed 2 minutes. Raise speed to high; beat until mixture is pale and thick, about 4 minutes more.

Sift cocoa mixture over egg mixture; carefully fold in with a large rubber spatula. When almost incorporated, pour melted butter down side of bowl; gently fold to combine. Pour batter into prepared sheet, and smooth top with an offset spatula.

Bake until cake springs back when touched, 9 to 10 minutes. Place a clean kitchen towel on a work surface; dust generously with cocoa. Remove pan from oven and immediately turn cake out onto a wire rack lined with parchment. Invert onto prepared towel; remove parchment.

While cake is still warm, gently roll, starting from one short side, into a log, incorporating towel. Let rolled cake cool completely. (This will give it a “shape memory,” so it will be easier to roll again with filling.)

Unroll cooled cake and transfer to parchment. Using a pastry brush, coat cake with rum. Using offset spatula, spread mousse over cake. Sprinkle with 1/2 cup reserved grated coconut.

With a short end of cake facing you, roll carefully into a roulade. Wrap in a kitchen towel; secure with clothespins or large metal binder clips to hold roll in a tight round shape. Place on a baking sheet and refrigerate until firm, at least 2 hours and up to 1 day.

Remove from refrigerator and unwrap. Use a serrated knife to trim ends of roll, cutting on the bias (starting 1 inch in from end on shorter side, and finishing 2 inches in, to create a wedge). Choose the more presentable trimmed end and place, cut-side up, on log, forming a branch. If desired, use other trimmed end for a second branch.

Transfer log to a serving platter. Slide strips of parchment under perimeter of log (to keep platter clean). Use offset spatula to coat log with frosting, then attach one or both trimmed ends; coat with frosting. Remove parchment pieces. Decorate with meringue mushrooms, coconut shavings, and rosemary sprigs.