1. Make the coconut sponge: Preheat the oven to 350 degrees F. Spray the bottom and sides of a 17-1/2 x 12-1/2 inch baking sheet with nonstick cooking spray. Line the bottom of the pan with parchment paper.
2. Fill a medium saucepan one-third full with water and bring to a simmer. Whisk together the eggs and sugar in the bowl of an electric mixer. Place the bowl over the pan of simmering water and whisk constantly until the egg mixture is warm to the touch. Transfer the bowl to the mixer stand and beat on high speed until it has tripled in volume, about 5 minutes. Using a rubber spatula, fold in the coconut just until blended. Pour the batter onto the prepared baking sheet and spread it evenly in the pan with a rubber spatula.
3. Bake for 20 to 25 minutes, until the top of the cake is light golden brown and a toothpick inserted in the center comes out clean. Cool the cake in the pan on a wire rack for 15 minutes.
4. Run a small sharp knife around the sides of the pan to loosen the cake. Place a wire rack over the cake and invert. Carefully peel off the parchment paper (the cake is extremely delicate). Cool the cake completely.
5. Make the ganache: Put the bittersweet and milk chocolates in a large bowl and set aside. Bring the cream to a boil in a medium saucepan. Immediately pour the hot cream over the chocolate. Whisk until the chocolate is completely melted and smooth. Cover the ganache with plastic wrap, pressing it directly against the surface, and refrigerate until it is firm enough to pipe, about 4 hours.
6. Assemble the cake: Trim off any uneven edges and cut the cake crosswise into three equal rectangles, each measuring about 5 X 10 inches. Place one of the rectangles on a serving platter. Using a small metal offset spatula, spread a generous layer of ganache over the top of the cake layer. Cover with another cake layer and spread the remaining ganache over the top and sides of the cake. Sprinkle the toasted coconut over the top and sides of the cake. If not serving immediately, refrigerate the cake. The cake can be made up to 1 day ahead. Bring to room temperature before serving.
MAKES 6 TO 8 SERVINGS