Preheat oven to 350 degrees F (175 degrees C). Grease and flour two 9-inch cake pans or one 9x13-inch pan.

Beat 2 cups sugar and shortening with an electric mixer in a large bowl until light and fluffy. Add the egg yolks one at a time, allowing each yolk to blend into the butter mixture before adding the next. Mix in the vanilla.

Place coconut in a food processor and pulse until finely chopped. Whisk coconut together with flour and baking powder; set aside. In a separate bowl, beat the egg whites until medium-stiff peaks form.

Pour the flour mixture into the batter alternately with the milk, mixing until just incorporated. Using a whisk or rubber spatula, gently fold 1/3 of the egg white mixture into the batter to lighten. Fold in remaining egg whites. Pour batter into prepared pans.

Bake in preheated oven until cake springs back when lightly touched with a finger and a tester inserted in the center comes out clean, about 30 minutes for layers or 40 minutes for a 9x13-inch pan. Let cakes cool completely on wire racks.

To Make White Frosting: shake 6 tablespoons flour and 1 cup milk in a jar. Transfer to a saucepan. Cook mixture over medium heat, stirring constantly, until mixture has thickened, about 15 minutes. Remove from heat and allow to cool.

Beat 1 cup sugar, butter, and 1/2 cup shortening with an electric mixer until smooth and fluffy. Add cooled milk mixture and beat well; stir in 1 teaspoon vanilla.