Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9x13-inch pan.

Melt butter in a large saucepan. Remove saucepan from heat and stir white sugar, eggs, and vanilla extract into the melted butter until thoroughly combined.

Mix flour, cocoa powder, salt, and baking powder together in a large mixing bowl; add butter mixture to the flour mixture and stir to form a smooth batter. Pour the batter into the prepared pan.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, 25 to 30 minutes. Cool completely.

Beat coconut, sweetened condensed milk, and confectioners' sugar together in the bowl of a stand mixer until very thick; spread over the cake in an even layer.

Heat prepared frosting in the microwave until the frosting has a pourable consistency, about 75 seconds; pour over the cake and spread into an even layer. Refrigerate cake until the frosting is set, at least 15 minutes.