Step 1: To make the cake: Butter two 9-inch cake pans, then line the bottoms with rounds of parchment or wax paper. Preheat the oven to 350°.
Step 2. In the bowl of a stand mixer, by hand, whisk the cocoa and boiling water until smooth. Cover with plastic wrap to prevent evaporation and cool to room temperature, about 30 minutes.
Step 3. Add the oil and yolks to the mixer bowl with the chocolate. Attach the whisk beater. Starting on low speed, gradually raise the speed to medium and beat for about 1 minute, or until smooth and shiny, and resembling a buttercream. Scrape down the sides of the bowl. Beat in vanilla for a few seconds.
Step 4. In a medium bowl, sift together the cake and all-purpose flour, sugar, baking powder, baking soda, and salt.
Step 5. Mix in half of the dry ingredients into the chocolate mixture and beat on low speed until the dry ingredients are moistened. Scrape down the sides of the bowl. Repeat with the remaining flour mixture. Raise the speed to medium high and beat for 1 minute. Scrape down the sides and the bottom of the bowl. The mixture will be very thick.
Step 6. On low speed add the egg whites. Gradually raise the speed to medium and beat for 2 minutes. The batter will be the consistency of thick soup.
Step 7. Divide the batter into the 2 prepared cake pans (each one will be 1/4 full), smooth the tops, and bake for about 25 to 35 minutes, until a toothpick inserted into the center comes out clean.
Important Note: During the baking, the batter will rise almost to the top of the pans and a little higher in the middle. They will start to lower just before the end of baking. To prevent the collapse of the delicate foam structure, while still hot the cakes must be unfolded as soon as they have baked.
Step 8: Leave oven on to toast the pecans.
Step 9: Cool cake layers completely. While the cakes are baking and cooling, make the filling and icing.
Step 10: To toast the pecans: Spread the pecans evenly on a baking sheet and bake for about 7 minutes to enhance the flavor. Stir once or twice to ensure even toasting and avoid overbrowning. Set aside.
Step 11: To make the filling: Mix the cream, sugar, and egg yolks in a medium saucepan. Put the 3 ounces butter, salt, toasted coconut, and pecan pieces in a large bowl.
Step 12. Heat the cream mixture and cook, stirring constantly (scraping the bottom as you stir) until the mixture begins to thicken and coats the spoon (an instant-read thermometer will read 170°.)
Step 13. Pour the hot custard immediately into the pecan-coconut mixture and stir until the butter is melted. Cool completely to room temperature. (It will thicken.)
Step 14: To make the syrup: In a small saucepan, heat the sugar and water until the sugar has melted. Remove from heat and stir in the dark rum.
Step 15: To make the icing: Process the 8 ounces of chopped chocolate in a food processor until fine.
Step 16: Heat the cream until it just begins to boil. Remove from heat and pour the cream through the feed tube of the food processor in a steady stream. Process for a few seconds until smooth.
Step 17:Pulse in the vanilla the chocolate.
Step 18: Cover and cool in the refrigerator. Stir every 20 to 30 minutes until it reaches a soft frosting consistency.
Step 19: To assemble the cake: Remove the cake layers from the pans and cut both cake layers in half horizontally, using a serrated bread knife.
Set the first cake layer on a cake plate. Brush well with syrup. Spread ¾ cup of the coconut filling over the cake layer, making sure to reach to the edges. Set another cake layer on top.
Step 20: Repeat, using the syrup to brush each cake layer, then spreading ¾ cup of the coconut filling over each layer, including the top.
Step 21. Ice the sides with the chocolate icing, then pipe a decorative border of chocolate icing around the top, encircling the coconut topping.