Make the Cake: Preheat oven to 350 degrees F. Grease and flour a 10-cup or 12-cup Bundt pan.
In a medium bowl, whisk together the flour, baking powder and salt.
Using an electric mixer, beat the butter and sugar on medium speed until light and fluffy, about 3 minutes. Scrape down the sides of the bowl, then add the eggs and the yolks one at a time, beating well after each addition. Scrape down the bowl again, add the coconut and vanilla extracts, and beat until just incorporated.
Add the flour mixture in three parts, alternating with the coconut milk, beginning and ending with the flour mixture, mixing after each addition until just combined; do not overmix.
Spoon the batter into the prepared pan, smoothing the top into an even layer. Bake until a sharp knife or toothpick inserted in the center of the cake comes out with just a few moist crumbs, 50 to 55 minutes. Transfer the pan to a wire rack to cool completely. Gently loosen the sides of the cake from the pan and turn it out onto the rack.
Make the Glaze: In a large bowl, whisk together 4 tablespoons of the coconut milk, the coconut extract and vanilla extract. Add the powdered sugar and whisk until incorporated and smooth. Slowly stir in the white chocolate. The glaze should be thick yet pourable; if the glaze appears too thick, thin it out with addition coconut milk, a tablespoon at a time, until you reach the desired consistency.
Pour the glaze in large, thick ribbons over the crown of the cake, allowing the glaze to spread and drip down the sides of the cake. Top with the toasted shredded coconut immediately and allow the glaze to set before serving, about 5 minutes. The cake can be kept in an airtight container at room temperature for up to 3 days.
Notes
Note: I used canned coconut milk for this recipe (Goya brand found in the Hispanic section of the local grocery store) - one 13.5-ounce can was enough for the cake and frosting.Nutritional values are based on one serving