Preheat the oven to 350 degrees F (175 degrees C). Line the bottom of a 15x10x1-inch pan with parchment paper.

Sift cake flour, baking powder, and salt together 3 times in a bowl.

Beat egg whites using an electric mixer in a mixing bowl until foamy. Add 1/2 of the total amount of sugar gradually; continue to beat until soft peaks form.

Beat butter using an electric mixer in a separate bowl until creamy. Gradually add remaining sugar; beat until light and fluffy. Add flour mixture alternately with coconut milk and almond milk in small amounts at a time, beating after each addition until smooth. Mix in flaked coconut, almonds, almond extract, coconut extract, and vanilla extract. Add egg white-sugar mixture and beat thoroughly into batter. Spread batter into the prepared pan.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, 30 to 35 minutes. Let cool in the pan for 10 minutes; remove cake and transfer to a wire rack to cool completely.

While cake continues to cool, beat butter, salt, vanilla extract, almond extract, and coconut extract together in a bowl using an electric mixer on medium speed until fluffy. Add powdered sugar to butter mixture alternately with coconut milk, beating until smooth. Frost cooled cake.