To prepare the coconut:

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. This may be done a week or two ahead. Save enough coconut (you’ll need about 1/2 cup finely grated, the smaller pieces are fine to save) for the filling. They may be wrapped well and frozen if the coconut is being done ahead, but do not grate until you are ready to fill the cake, then grate them on the small holes of a box grater. You will have extra coconut.

To prepare cake:

Heat oven to 400 degrees. Line a 12-by-17-inch jellyroll pan with parchment paper, and set aside.

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.

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. (I use the same bowl and whisk.)

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 the cake out onto a wire rack lined with parchment paper. Peel parchment paper from the top of the cake, and cool completely.

To prepare White Chocolate Mousse:

Dissolve gelatin in 1/4 cup cold water, and set aside for 5 minutes. Place chocolate in the bowl of a food processor, and pulse until very finely chopped.

Prepare an ice-water bath; set aside. Place 3/4 cup cream in a small saucepan, and bring just to a boil over medium-high heat. Add gelatin, and stir for 30 seconds to dissolve completely. Pour into food processor with the motor running, and process until chocolate mixture is smooth.

Transfer to a medium bowl and place over ice water bath. Chill until mixture is thick enough to fall from a spoon in ribbons.Whip remaining 1 3/4 cups heavy cream to soft peaks. Fold into chocolate mixture. If not using immediately, refrigerate in an airtight container.

To prepare the Seven Minute Frosting:

In a small heavy saucepan, combine 3/4 cup sugar, the water, and corn syrup; cook, stirring occasionally, over medium heat, until sugar dissolves. Raise heat, and bring to a boil, washing down sides of pan with a pastry brush dipped in water to prevent crystals from forming. Continue cooking without stirring until a candy thermometer registers 230 degrees, about 5 minutes.

Meanwhile, beat egg whites in the bowl of an electric mixer fitted with the whisk attachment on medium speed until soft peaks form, about 2 1/2 minutes. Gradually add remaining tablespoon sugar until combined.

Once sugar syrup has reached 230 degrees, immediately remove from heat; with mixer on medium-low, carefully pour mixture into egg-white mixture in a steady stream down the sides of the bowl to prevent splattering. Increase speed to medium-high, and beat until mixture is completely cool and stiff peaks form, about 7 minutes. Add vanilla, and beat until fully incorporated. Frosting should be shiny and smooth. Use immediately.

To assemble the cake:

Place a thin, smooth cotton or linen 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 long 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, and coconut shavings and greenery dusted in sugar if desired.

To Prepare the Meringue Mushrooms:

Heat oven to 200 degrees. Place Swiss Meringue (below) in an 18-inch pastry bag fitted with a large round tip (Wilton #1A). Line two baking sheets with parchment. Pipe meringue, forming domes 1 to 2 inches in diameter. Pipe a stem shape for each dome. Place baking sheets in oven for 1 hour, reduce heat to 175 degrees, and continue baking until meringues are completely dry to the touch but not browned, 45 to 60 minutes.

Fill a medium saucepan one-quarter full with water. Set over medium heat, and bring water to a simmer. Place bittersweet chocolate in a heat-proof bowl, and set over simmering water. Turn off heat; stir occasionally until completely melted. Using an offset spatula, spread bottoms of domed meringues with chocolate; let set at room temperature.

Place white chocolate in a heat-proof bowl, and set over simmering water. Turn off heat; stir occasionally until completely melted. Allow to cool slightly. Spread white chocolate over dark chocolate; use a toothpick to create lines from the center of the cap to the edges. Let set in a cool, dry place.

Poke a small hole in the center of each mushroom cap using a paring knife. Dip the pointed end of each of the mushroom stems in melted white chocolate, and insert into hole in center of cap. Allow to set. Meringue mushrooms should be kept in an airtight container in a cool, dry place.

Very lightly dust the tops of the meringue mushrooms, with a fine sieve, with a little cocoa powder.

To Prepare the Swiss Meringue for the Mushrooms:

Fill a medium saucepan one-quarter full with water. Set over medium heat, and bring water to a simmer.
Combine the egg whites, sugar, and cream of tartar in the bowl of an electric mixer, and place over saucepan. Whisk constantly until sugar is dissolved and whites are warm to the touch, about 3 minutes. Rub the mixture between your fingers to test.

Transfer bowl to mixer stand, and whip, starting on low speed and gradually increasing to high, until meringue is cool and stiff glossy peaks form, about 10 minutes. Add vanilla; mix until combined. Use immediately.