Fresh Coconut Layer Cake

Name: Fresh Coconut Layer Cake
Cuisine: American
Ingredients: 1 ¼ cups heavy whipping cream; ¾ cup white sugar; 2 tablespoons cream of coconut; 1 tablespoon cornstarch; 1 tablespoon coconut milk; ½ cup unsalted butter, softened; 1 cup grated fresh coconut; 3 cups all-purpose flour; 1 tablespoon baking powder; ¼ teaspoon salt; 1 ½ cups white sugar; 4 eggs; ¾ cup unsalted butter, cubed; 1 tablespoon vanilla extract; 1 cup coconut milk; 6 ounces cream cheese, softened; 4 cups sifted confectioners' sugar; 3 tablespoons coconut milk; 1 cup white sugar; 1 cup water; ¼ teaspoon vanilla extract; ½ cup shredded coconut