Preheat oven to 350°F (180°C). Spray 2 (3-inch-tall) 8-inch round cake pans (see Note) with baking spray with flour.
For milk chocolate cake layer: In a medium bowl, place melted butter. Whisk in sugars, eggs, and vanilla.
In another medium bowl, whisk together flour, unsweetened cocoa, baking powder, baking soda, and salt. Gradually add flour mixture to butter mixture alternately with buttermilk, beginning and ending with flour mixture, beating just until combined after each addition. Stir in melted milk chocolate until combined. Pour into 1 prepared pan.
For dark chocolate cake layer: In a medium bowl, place melted butter. Whisk in sugars, eggs, and vanilla.
In another medium bowl, whisk together flour, Dutch process cocoa, baking powder, baking soda, and salt. Gradually add flour mixture to butter mixture alternately with buttermilk, beginning and ending with flour mixture, beating just until combined after each addition. Stir in melted bittersweet chocolate until combined. Pour into remaining prepared pan.
Bake until a wooden pick inserted in center comes out clean, 40 to 45 minutes. Let cool in pans for 15 minutes. Remove from pans, and let cool completely on wire racks.
Place dark chocolate cake layer on a cake stand. Spread 1½ cups White Chocolate Buttercream on top. Place milk chocolate cake layer on top of buttercream. Spread remaining White Chocolate Buttercream on top and sides of cake. Garnish with chopped chocolate, if desired.