Preheat oven to 350 degrees F. Grease the bottom and 1/2 inch up the sides of a 9x5x3-inch loaf pan. Line bottom of pan with waxed paper or parchment paper; grease paper. Set pan aside. In a large bowl stir together flour, sugar, cocoa powder, baking powder, and salt. Make a well in the center of flour mixture; set aside.
In a medium bowl combine egg, milk, and oil. Add egg mixture all at once to flour mixture. Stir just until moistened (batter should be lumpy). Fold in white baking pieces, 1/3 cup dark chocolate pieces, and 1/3 cup milk chocolate pieces. Spoon batter into the prepared loaf pan, spreading evenly.
Bake for 45 to 55 minutes or until a wooden toothpick inserted in the center comes out clean. Cool in pan on a wire rack for 10 minutes. Remove from pan; peel off paper. Cool completely on wire rack. Wrap and store overnight before slicing.
If desired, top each serving with sweetened whipped cream and drizzle with melted chocolate.