In a large bowl with an electric mixer, beat the brown sugar, butter and oil until well blended. Add the egg yolks 1 at a time and beat well after each addition. Add the vanilla extract.
In a separate bowl, combine the flour, baking powder, baking soda and salt. Add one-quarter of the dry ingredients to the sugar-butter mixture followed by one-quarter of the yogurt, beating well after each addition. Then add half of the remaining dry ingredients followed by the remaining yogurt, beating well after each addition. Beat in the remaining dry ingredients and stir in the coconut.
In a separate bowl with an electric mixer and the whisk attachment, beat the egg whites until stiff peaks form. Fold half of the egg whites into the batter, then fold in the remaining whites.
Grease and flour three 9-inch round cake pans. Divide the batter among the pans. Bake until a toothpick inserted in the center comes out clean, about 30 minutes. Let cool for 10 minutes in the pans, then remove from the pans onto a rack to cool completely.
For the cream cheese frosting: In a bowl with an electric mixer, beat the cream cheese and butter until well incorporated. Add the extract and mix until incorporated. Add the powdered sugar 1 cup at a time and beat until smooth.
Place one of the cake layers on a serving plate. Spread about a third of the frosting onto the cake layer and generously sprinkle some toasted coconut on top. Repeat with the second layer. Top with the final layer and spread the remaining frosting over the top and sides of the cake. Sprinkle with remaining toasted coconut. Keep refrigerated until serving.
This recipe was created by a contestant during a cooking competition. It has not been tested for home use.