Preheat oven to 350 degrees F (175 degrees C). Grease 2 baking pans.

Place brown sugar, 1/4 cup butter, and coconut flakes in a microwave-safe container and heat until warm, about 2 minutes. Mix into a paste.

Combine flour, baking powder, and salt in a separate bowl. Whip egg whites in a metal bowl with a clean electric mixer until stiff peaks are formed.

Beat 1/2 cup white sugar and 7 1/3 tablespoons butter in a bowl with an electric mixer until creamy. Add remaining 1/2 cup white sugar, cream of coconut, reserved pineapple juice, egg yolks, and vanilla extract; mix thoroughly. Add the flour mixture; stir batter slowly until fully combined. Fold in the whipped egg whites using a spatula.

Spread the brown sugar paste over the bottoms of the prepared pans. Arrange pineapple chunks on top. Pour in the batter.

Bake in the preheated oven until tops are brown and crisp and a toothpick inserted into the centers comes out clean, about 1 hour. Let cool, about 25 minutes. Flip cakes over onto a serving plate and out of the baking pans.