Position rack in center of oven; preheat to 350 degrees. Butter two 5-inch cake pans with 2-inch high sides. Line bottom of pans with parchment paper; butter parchment. Combine cocoa powder and milk chocolate in medium bowl. Pour 1/4 cup boiling water over; whisk until mixture is smooth. Whisk in buttermilk.
Whisk flour, baking soda, and salt in another medium bowl. Using electric mixer, beat both sugars, oil, egg, and vanilla in large bowl until well blended. Add flour and cocoa mixtures; beat until blended (batter will be thin). Divide batter between pans.
Bake cakes until tester inserted into center comes out with some crumbs attached, 28-30 minutes. Cool in pans on rack for 15 minutes. Turn cake out onto rack; peel off parchment. Turn over; cool on racks.
Place chocolate in large metal bowl. Set bowl over saucepan of simmering water and stir until melted and smooth. Remove bowl from over water. Add butter and stir until melted, then add sour cream, and corn syrup and whisk until smooth. Let frosting stand at room temperature until thick enough to spread, about 20 minutes.
Using serrated knife, trim top of cakes to make level. Cut each cake horizontally in half. Place 1 cake layer, cut side up, on platter. Spread 1/4 cup frosting over, leaving 1/2-inch border. Top with second layer, cut side down. Spread 1/4 cup frosting over leaving 1/2-inch border. Top with third layer, cut side up. Spread 1/4 cup frosting over leaving 1/2-inch border. Top with remaining layer, cut side down. Spread 1/3 cup frosting over top and side. Chill until frosting is set, about 30 minutes. Keep remaining frosting at room temperature.
Spread remaining frosting evenly over top and side of cake. Can be made 2 days in advance. Cover with cake dome and store at room temperature.