Preheat the oven to 350°F. Grease 2 or 3 (9-inch) round cake pans with butter or nonstick cooking spray. Line the bottom with parchment paper and grease. Lightly flour the pans; set aside.

In a large bowl using an electric mixer on medium speed, cream the butter and sugar together until light and fluffy. Add the eggs, one at a time, beating well after each before adding the next one.

Sift the flour one more time, adding the baking powder. Add it in thirds to the batter, alternating with the coconut milk and beating well after each addition. Beat in the extracts.

Divide the batter evenly between the cake pans, and gently tap the bottoms against the countertop to get rid of any air bubbles. Bake until a cake tester inserted in the center comes out clean, about 30 minutes (start testing at 21 minutes for 3 pans). Remove from the oven and let the cakes cool in the pans at least 10 minutes, then invert the pans onto greased wire racks. Remove and discard the parchment paper, then turn the cakes over and let them cool completely before frosting.

Set one layer on a cake plate and cover evenly with a thin layer of the simple syrup followed by 3⁄4 to 1 cup of the frosting. Set the other layer on top and cover with the rest of the simple syrup and coconut shreds (from the syrup), then use the remaining frosting to ice the top and side of the cake. Cover the top and side of the cake evenly with the coconut flakes. This cake does not require refrigeration.

For the coconut simple syrup: Combine the water, sugar, and coconut in a small skillet over medium heat. Stir to dissolve the sugar and bring to a boil, then turn the heat off. Let cool completely before using, may be stored in an airtight container for several days. If you don't like the texture of shredded coconut, you can strain the syrup before using, but we like the coconut flakes between the layers of the cake.

For the icing: Whisk the sugar, corn syrup, cream of tartar, salt, and water together in a small, heavy-bottomed saucepan. Bring to a boil over medium heat (do not stir it) and cook, undisturbed, until it reaches 240°F on a candy thermometer, about 2 minutes; stirring can cause the sugar to crystallize.

Meanwhile, in a medium bowl using an electric mixer on high speed, beat the egg whites until foamy.

Once the sugar syrup reaches 240°F, immediately and carefully pour it into the egg whites, with the mixer running on high speed, away from the beaters to prevent splattering and possible burns. Continue to beat until the frosting is glossy and fluffy, about 5 minutes. Beat in the coconut extract. Use immediately.