Green food coloring, optional
To prepare the cake: Preheat the oven to 375 degrees. Lightly oil 3 (8-inch) pans, line the bottoms with parchment paper and lightly oil the paper.
Whirl the coconut in a food processor until it is the texture of coarse meal. Add the baking powder and whirl briefly, to mix. In the large bowl of the electric mixer, beat the egg whites until they are frothy, then add the sugar in a slow stream, and beat at high speed several minutes, scraping the bowl occasionally, until the mixture is thickened and is starting to get some body. In the small mixer bowl (no need to clean beaters), whip the egg yolks briefly.
Gently fold the yolks into the whites, and when they are almost fully mixed, gently fold in the coconut mixture. Divide evenly among the three pans. Bake about 20 minutes. The tops should be firm. Cool 10 minutes, run a knife around the edge, invert and peel away parchment paper. Cool completely.
Meanwhile, make the mousse: Grate the zest of the limes and set aside. Juice the limes into a small bowl. Sprinkle the gelatin onto the lime juice. Allow to thicken for 5 minutes, then add1/4cup of boiling water, stirring well to dissolve the gelatin. Set aside.