1¼ cups hot water
2/3 cup sour cream
2 2/3 cups all purpose flour
2 teaspoons baking powder
1 teaspoon baking soda
½ teaspoon salt
1½ sticks softened unsalted butter
1½ cups sugar
½ cup softened solid vegetable shortening
¾ cup natural unsweetened cocoa powder
1 cup (packed) dark brown sugar
3 large eggs
1 tablespoon vanilla extract
5 large egg whites
1 2/3 cups sugar, divided
1 vanilla bean, split lengthwise
1/3 cup water
4 sticks softened unsalted butter, diced
5 ounces high-quality milk chocolate chopped
5 ounces bittersweet chocolate (54% to 60% cacao), chopped
1¼ cups heavy whipping cream
2 tablespoons light corn syrup
¼ stick softened unsalted butter, in ½" cubes
Butter three 8-inch-diameter cake pans with 1½"-high sides. Line bottom of each pan with parchment paper round; butter parchment. Dust pans with flour; tap out excess. Whisk 1¼ ups hot water, cocoa powder, and sour cream in medium bowl until smooth; set aside until mixture is cool, 20 to 30 minutes.
Position rack in center of oven and preheat to 325°F. Sift next 4 ingredients into another medium bowl. Using electric mixer, beat butter and shortening in large bowl until well blended and smooth. Add both sugars; beat until light and fluffy, about 5 minutes. Add eggs 1 at a time, beating until well blended after each addition. Mix in vanilla. Add flour mixture in 3 additions alternately with cocoa mixture in 2 additions, beginning and ending with flour mixture and beating just until incorporated. Divide batter among prepared pans (about 2 3/4 cups batter for each) and smooth tops.
Bake cakes until tester inserted into center comes out clean, 50 to 55 minutes. Cool cakes in pans on racks 20 minutes. Run knife around sides of cakes to loosen. Invert cakes onto racks and cool completely. DO AHEAD Can be made 1 day ahead. Wrap in plastic and store at room temperature.
Combine egg whites and 1/3 cup sugar in large bowl of stand mixer fitted with whisk attachment; scrape in seeds from vanilla bean.
Combine remaining 1 1/3 cups sugar and 1/3 cup water in medium saucepan. Stir over medium heat until sugar dissolves. Attach candy thermometer to side of pan, making sure that bulb of thermometer is immersed in syrup. Increase heat; boil until temperature registers 238°F to 240°F (soft-ball stage), about 5 minutes. Immediately remove syrup from heat (use within 2 minutes).
Meanwhile, beat egg white mixture on medium speed until very soft peaks form (mixture will be slightly opaque).
Increase mixer speed to high and slowly pour hot syrup down side of bowl into egg white mixture in slow steady stream; beat until meringue forms stiff peaks. Cool meringue in bowl until lukewarm (instant-read thermometer will register 100°F), about 30 minutes. Do not beat.
Start beating meringue again on medium speed of stand mixer. Gradually add butter, 2 to 3 tablespoons at a time, beating constantly until absorbed before adding next addition. Continue beating until buttercream is smooth. (If buttercream looks broken or curdled, place mixer bowl with buttercream over medium heat on stove burner and whisk with large whisk for 5 to 10 seconds to warm mixture slightly, then remove from heat and whisk vigorously or attach bowl to stand mixer and beat mixture again on medium speed. Repeat warming and beating buttercream mixture as many times as needed until buttercream is smooth and no longer curdled.)
Using serrated knife, cut each cake in half horizontally, forming 6 cake layers. If necessary, trim any domed tops of cakes to form even layers. Place 1 cake layer on platter. Drop 3/4 cup buttercream by tablespoonfuls over top of cake layer, then spread evenly to edges with offset spatula. Top with second cake layer, then spread ¾ cup buttercream over. Repeat with 3 more cake layers, spreading ¾ cup buttercream over each and pressing slightly to adhere. Top with 6th cake layer (do not spread buttercream over top cake layer). Chill cake at least 1 hour.
Combine both chocolates in medium bowl. Bring cream and corn syrup to simmer in heavy saucepan. Pour hot cream mixture over chocolate; let stand 2 minutes. Whisk chocolate mixture until melted and smooth. Add butter; whisk until melted. Chill glaze until slightly thickened and glaze drips thickly when poured slowly from spoon, 30 to 45 minutes.
Spoon glaze around top edge of cake by teaspoonfuls, spacing drips up to 2 inches apart and allowing glaze to drip slowly down sides of cake. Spoon remaining glaze over top center of cake and smooth with offset spatula, covering top completely. Chill cake until glaze sets, at least 1 hour.