Coconut Pastry Cream

3/4 cup whole milk
3/4 cup unsweetened coconut milk
1 teaspoon vanilla bean paste
4 large egg yolks
1/3 cup granulated sugar
4 tablespoons cornstarch
1/2 teaspoon pure vanilla extract 
1/2 teaspoon coconut extract
1/4 cup cold heavy cream

Combine the milks and vanilla bean paste in a medium nonreactive saucepan and bring to a simmer over low heat.
Whisk together the yolks, sugar and cornstarch in a large bowl. Slowly whisk the warm milk into the egg mixture then return the mixture to the pot over medium heat and bring to a boil, whisking constantly, until thickened. Scrape the mixture into a bowl and whisk in the vanilla and coconut extracts.  Let cool to room temperature then cover with plastic wrap and refrigerate until cold, at least 2 hours.

When ready to assemble the cake, place cold pastry cream in to a large mixing bowl.  Add cold heavy cream and beat using a handheld mixer until cream is whipped and fluffy.

Coconut Simple Syrup (adapted from Bobby Flay)

1 cup water
1 tablespoon sugar
3/4 cup sweetened flaked coconut

Bring water and sugar to a boil. Stir in the coconut, remove from the heat and let sit for at least 30 minutes and up to 4 hours. Strain the liquid into a clean saucepan, bring to a boil and let cook until the mixture is slightly reduced, about 5 minutes. Let cool.
Coconut Cake (adapted from Cook's Illustrated) 

1 large egg
5 large egg whites
3/4 cup cream of coconut (the kind you use for pina coladas)
1/4 cup water
1 1/2 teaspoons coconut extract
2 1/4 cups cake flour (9 ounces), sifted
1 cup granulated sugar
1 tablespoon baking powder
3/4 teaspoon table salt
12 tablespoons unsalted butter (1 1/2 sticks), cut into 12 pieces, softened, but still cool   
2 cups packed sweetened flaked coconut (about 8 ounces)
Adjust oven rack to lower-middle position and heat oven to 325 degrees. Grease two 9-inch round cake pans with shortening and dust with flour.
Beat egg whites and whole egg in large measuring cup with fork to combine. Add cream of coconut, water, and coconut extract and beat with fork until thoroughly combined.

Combine flour, sugar, baking powder, and salt in bowl of standing mixer fitted with paddle attachment. Mix on lowest speed to combine, about 30 seconds. With mixer still running on lowest speed, add butter 1 piece at a time, then beat until mixture resembles coarse meal, with butter bits no larger than small peas, 2 to 2 1/2 minutes.
With mixer still running, add 1 cup liquid. Increase speed to medium-high and beat until light and fluffy, about 45 seconds. With mixer still running, add remaining 1 cup liquid in steady stream (this should take about 15 seconds). Stop mixer and scrape down bowl with rubber spatula, then beat at medium-high speed to combine, about 15 seconds. (Batter will be thick.)
Divide batter between cake pans and level with offset or rubber spatula. Bake until deep golden brown, cakes pull away from sides of pans, and toothpick inserted into center of cakes comes out clean, about 30 minutes (rotate cakes after about 20 minutes). Do not turn off oven.
Cool in pans on wire racks about 10 minutes, then loosen cakes from sides of pans with paring knife, invert cakes onto racks and then re-invert; cool to room temperature.  
While cakes are cooling, spread shredded coconut on rimmed baking sheet; toast in oven until shreds are a mix of golden brown and white, about 15 to 20 minutes, stirring 2 or 3 times. Cool to room temperature.