Preheat the oven to 350°F. Butter and flour 3 (9-inch) round cake pans or spray with baking spray. Line the bottom of each pan with a circle of parchment paper.

In a medium bowl, whisk to combine the flour, baking powder, baking soda, and salt. Set aside.

In a large mixing bowl or the bowl of a stand mixer with the paddle attachment, combine sugar, butter, and vegetable oil. Beat on medium speed for about 3 minutes, until light and creamy. Add the egg whites, vanilla, and coconut extract and mix until combined, about 1 minute.

With the mixer on low speed, add about one-third of the flour mixture. Mix until completely combined. Add ½ of the coconut milk and mix until combined. Repeat adding another one-third of the flour mixture, remaining coconut milk mixture, then all of the remaining flour mixture. Stop and scrape down the bowl at least once during mixing, then mix until completely combined.

Fold in the shredded coconut. Divide the batter evenly among the cake pans and smooth out to the sides with a spatula (the batter will be thick).

Bake for 30 to 32 minutes, rotating the pans after 20 minutes, or until the center of the cake springs back when gently pressed.

Let the cakes cool in the pan for 20 minutes. Carefully invert, remove the parchment paper, and continue cooling on a wire rack.

In a large mixing bowl or the bowl of a stand mixer with the paddle attachment, beat the butter and cream cheese on medium speed until creamy and fluffy, about 3 minutes. Add the vanilla extract, coconut extract, and salt. Beat until combined. Scrape down the sides and bottom of the bowl.

Add the sugar 1 cup at a time, mixing on low speed between each addition. Once combined, scrape down the bowl, then mix on medium speed until fluffy, about 1 minute.

Place one cake layer on a cake stand. Spread about 1 cup of frosting evenly over the top. Top with another cake layer and repeat with more frosting. Top with the remaining cake layer. Spread the remaining frosting all over the outside of the cake. Press the coconut to the frosting all over the top and sides of the cake to cover completely. Chill for 30 minutes for easy slicing, or keep refrigerated for up to 3 days or until ready to serve.