Coconut–Key Lime Sheet Cake

Name: Coconut–Key Lime Sheet Cake
Cuisine: American, Southern
Ingredients: 1 1/2 cups unsweetened shredded coconut; Nonstick vegetable oil spray; 1 3/4 cups cake flour, plus more for pan; 1/2 teaspoon baking powder; 1/2 teaspoon kosher salt; 1/3 cup virgin coconut oil, melted; 1 1/2 teaspoons vanilla extract; 1 1/3 cups canned unsweetened coconut milk, divided; 1 1/2 cups sugar; 3/4 cup (1 1/2 sticks) unsalted butter, room temperature; 3 large eggs; 1 tablespoon finely grated key lime zest; 1/3 cup fresh key lime juice (from about 6 key limes); 1/4 cup (1/2 stick) unsalted butter, cut into pieces, room temperature; 3 large egg yolks; 1/2 cup sugar; 1 teaspoon finely grated key lime zest; 1/4 cup fresh key lime juice (from about 4 key limes); 1 1/2 cups cold heavy cream; 1 cup unsweetened coconut flakes