Butter the sides of two 9-inch cake pans and line the bottoms of the pans with 9-inch parchment paper.
Beat unsalted butter and sugar in a large mixing bowl with an electric mixer until light and fluffy, about 5 minutes.
Stir flour, baking powder, and salt in a bowl; set aside.
Whisk egg whites, milk, and coconut extract in a separate bowl.
Thoroughly mix about 1/3 of the flour mixture into the creamed butter and sugar; stir in half the egg white-milk mixture. Repeat twice more, ending with final 1/3 of flour mixture; stir to make a smooth batter.
Pour batter into prepared cake pans and smooth tops with a spatula.
Bake in the preheated oven until lightly browned and a toothpick inserted into the center of a cake comes out clean, 25 to 30 minutes.
Cool cakes in pans on racks for about 5 minutes before turning out onto racks to cool completely. Peel parchment paper from cakes.
Stir sweetened condensed milk, 1/4 cup heavy cream, and vanilla extract together in a bowl; stir in 3/4 cup flaked coconut. Refrigerate until chilled, about 1 hour.
Bring 1/2 cup heavy cream to a simmer over low heat; gradually stir in chocolate until thoroughly melted and the mixture is thick and smooth. Allow to cool to lukewarm.
Place a cake layer onto a serving platter; top with the condensed milk-coconut mixture.
Place second cake layer over the coconut filling.
Spread warm chocolate mixture evenly over top and sides of cake with a thin spatula.
Toss 1 cup flaked coconut in a bowl with food coloring until coconut is evenly colored; sprinkle coconut over chocolate topping to serve.
