Preheat the oven to 350°. Butter four 9-inch cake pans. Line the pans with parchment paper and butter and flour the paper. In a standing electric mixer fitted with the whisk, combine the eggs, sugar and vanilla and beat at high speed until the mixture is light, fluffy and tripled in volume. With the mixer on, gradually beat in the oil until incorporated. In a medium bowl, sift the flour with the cornstarch and salt. Gently fold the dry ingredients into the beaten-egg mixture until thoroughly incorporated.
Pour the batter into the prepared pans and bake the cakes for 20 to 22 minutes, or until a toothpick inserted in the center comes out clean. Transfer the cakes to a rack and let cool. Unmold the cakes and peel off the parchment paper.
In a small saucepan, combine the sugar with the water and cook over high heat, stirring constantly, just until the sugar is completely dissolved. Let cool, then stir in the rum.
In a small saucepan, combine the sugar and water and stir over high heat until the sugar is completely dissolved. Stop stirring, then boil the syrup until it reaches 238° (soft ball stage) on a candy thermometer.
Meanwhile, in a standing electric mixer fitted with the whisk, beat the egg whites until firm peaks form. With the mixer at medium-high speed, gradually beat in the hot sugar syrup. Reduce the speed to medium and beat until the meringue is cooled to room temperature. Beat in the butter, 2 or 3 tablespoons at a time, until all of the butter has been incorporated. Beat in the coconut extract.
Brush the rum soak over the cakes. Set one layer on a large plate and spread 1 1/4 cups of the frosting on top; repeat with the next 2 cake layers. Cover with the final cake layer. Spread the remaining frosting evenly around the side and over the top of the cake, swirling it decoratively on top. Gently press the toasted coconut onto the side of the cake. Refrigerate the cake for at least 1 hour before serving.