Make the Coconut Sponge
Preheat your oven to 350°F (175°C). Spray a 17½ × 12½-inch rimmed baking sheet with cooking spray and line it with parchment paper.
Create a double boiler by filling a medium saucepan one-third full of water and bringing it to a simmer.
In the bowl of a stand mixer, whisk the sugar and eggs until pale yellow.
Place the bowl over the simmering water and continue whisking constantly until the mixture is warm to the touch.
Return the bowl to the mixer fitted with the whisk attachment.
Beat on high speed for about 5 minutes, until the mixture has doubled in volume and is light and airy.
Using a rubber spatula, gently fold in the shredded coconut until just combined. Be careful not to deflate the egg mixture.
Pour the batter onto the prepared baking sheet and spread it into an even layer.
Bake for 20 to 25 minutes, until the top is light golden brown and a toothpick inserted in the center comes out clean.
Remove the cake from the oven and let it cool on a wire rack for 15 minutes.
Run a small knife around the edges of the pan to loosen the cake.
Place another wire rack on top of the cake and carefully invert it. Gently peel off the parchment paper.
Let the cake cool completely.
Prepare the Ganache
Place the chopped bittersweet and milk chocolate in a large heatproof bowl.
In a medium saucepan, bring the heavy cream to a boil.
Immediately pour the hot cream over the chocolate.
Let it sit for a minute, then whisk until the chocolate is completely melted and the ganache is smooth and glossy.
Cover the ganache with plastic wrap, pressing it directly onto the surface to prevent a skin from forming.
Refrigerate for about 4 hours, or until it's firm enough to spread.
Assemble the Cake
Once the cake is cool, trim any uneven edges.
Cut the cake crosswise into three equal rectangles (each about 5 by 10 inches).
Place one cake rectangle on a serving platter.
Using an offset spatula, spread a generous layer of the chilled ganache over the top.
Place the second cake layer on top and repeat with another layer of ganache.
Top with the final cake layer.
Use the remaining ganache to frost the top and sides of the cake, creating a smooth or textured finish.
Garnish the cake by sprinkling the toasted coconut over the top and pressing it gently onto the sides.
If not serving immediately, refrigerate the cake. For the best flavor and texture, bring it to room temperature before serving.
