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1. Butter or coat a 9x13-inch sheet pan with cooking spray. You can fudge a little on these measurements. A 7x11 or 10x15, for example. But don't try to cram this quantity of batter into a 9-inch square or stretch it across a half-sheet pan.
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2. Preheat your oven to 375 F and make sure your rack is in the middle position.
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3. In a small bowl, stir together the all-purpose flour, white whole-wheat flour, coconut flour, baking powder, baking soda, and salt. Set aside.
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4. In a large saucepan, bring the butter and the water just to a boil, stirring once or twice while waiting for this to happen.
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5. As soon as the butter/water mixture boils, pull the pan off the heat. Dump the flour mixture into the pan, followed by the sugar, eggs, reduced-fat sour cream, applesauce and coconut extract. Stir just until incorporated.
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6. Pour the cake batter into the prepared pan, smooth the top, and bake for 15-20 minutes or until a toothpick inserted into the center of the cake emerges with moist crumbs clinging to it. Be careful not to over-bake this cake. You want the aforementioned moist crumbs-not wet batter, but not a totally clean pick, either.
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7. While your cake is baking, make the icing by melting the white sugar, butter, and milk in a medium-sized saucepan over medium heat. Bring the mixture to a boil, stirring constantly, then reduce the heat to medium-low and cook while stirring until the sugar is dissolved. Remove from heat and let it cool for 2-5 minutes while the cake finishes baking. If the icing base mixture starts to set before your cake comes out of the oven, put it back over medium heat and resume your stirring just long enough to liquefy it again.
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8. When your cake is done, remove it from the oven, poke the entire surface with a fork, and set it on top of the stove or on a wire rack while you finish off your icing. Add the confectioners' sugar and coconut extract into the cooked sugar mixture and beat with a wooden spoon until smooth. If the icing looks too thick to pour easily, add a little more milk, a few drops at a time, until you have a manageable but not runny consistency.
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9. Immediately pour the warm icing over your fork-poked cake and spread it evenly from edge to edge. Without delay, sprinkle the entire surface generously with flaked coconut.
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10. Enjoy warm or at room temperature. If you have any to store, be sure to cover your pan tightly with plastic wrap or foil so it doesn't dry out. Eating the entire cake the day it's made also solves this problem nicely.
