Place an oven rack in the bottom third of the oven and another in the top third of the oven. Preheat the oven to 350 degrees. Butter three 9-inch cake pans, then line each pan with a parchment paper round. Butter the paper and dust the pans with flour; knock out the excess.
CAKE: Using an electric mixer fitted with the paddle attachment, cream the butter and sugar on medium-high speed until light and fluffy, about 3 minutes. Use a rubber spatula to scrape down the sides of the bowl.
In a medium bowl, stir together the flour, baking powder, and salt to blend. In a small bowl, whisk together the milk, coconut milk, coconut cream and coconut extract, until smooth. Add the flour mixture in 3 increments, alternating with the milk mixture in 2 increments, starting and ending with the flour mixture. After each addition, mix at low speed just to combine the ingredients.
Using an electric mixer fitted with the whisk attachment, beat the egg whites on high speed until stiff peaks form. Gently fold the egg whites into the batter until evenly blended. Divide the cake batter evenly among the prepared cake pans.
Set two layers on the top rack and the third on the lower rack. Stagger the cake layers on the oven racks so that no layer is directly over another. Bake for 25-30 minutes, or until a toothpick inserted in the middle of the cakes comes out clean. Monitor the layers carefully for doneness; each one may be done at a different time. Remove from the oven and let cool in the pans for 10 minutes, then unmold onto wire racks to cool completely.
7-MINUTE COCONUT FROSTING: In the heatproof bowl of an electric mixer set over a saucepan of simmering water, combine sugar, corn syrup, water, and egg whites. Cook over medium heat, stirring constantly, until mixture registers 160 degrees on an instant-read thermometer, about 2 minutes.
Attach bowl to a mixer fitted with the whisk attachment. Beat mixture on high speed until glossy and voluminous, about 5 minutes. Beat in coconut extract. Frost cake immediately.
TO ASSEMBLE CAKE: Stack one cake layer on a serving plate, and spread the top with desired filling. Sprinkle with coconut. Repeat with the second layer. Stack the final cake layer on top of the first two, and cover the top and sides with the frosting. Sprinkle the remaining coconut on the top and sides of the cake.
Cover the cake loosely, store for 1 day at room temperature OR up to 3 days in the refrigerator. Bring to room temperature before serving.
Cook's Note: There is enough 7-minute frosting to use for filling, but may we suggest one of the follow fillings: Lemon or Lime Curd, Raspberry Jam, Coconut Custard, or Cream Cheese Frosting.