Step 1

Preheat oven to 350°F. Lightly butter three 9-inch cake pans with 1 3/4-inch-high sides. Line pan bottoms with waxed paper rounds. Butter paper.

Step 2

Bring water and coffee powder to boil in heavy small saucepan. Remove from heat. Add cocoa and whisk until smooth. Cool completely.

Step 3

Sift flour, baking powder, baking soda and salt into medium bowl. Using electric mixer, beat butter in large bowl until fluffy. Add sugar in 4 additions, beating well after each addition and scraping down sides of bowl occasionally. Add eggs and yolks 1 at a time, beating just to blend after each addition. Using rubber spatula, mix dry ingredients into butter mixture alternately with cocoa mixture, beginning and ending with dry ingredients. Divide batter evenly among prepared pans. Bake until toothpick inserted into centers come out clean, about 23 minutes. Cook cakes in pans on racks 10 minutes. Turn out onto racks. Peel off waxed paper; cool completely.

Step 4

DO AHEAD: Can be prepared 1 day ahead. Return cakes to pans and cover tightly with foil. Let stand at room temperature.

Step 5

Carefully scrape seeds from vanilla beans into heavy large saucepan (save beans for another use). Add cream, sugar, yolks, and salt and blend well. Stir over medium-low heat until custard thickens and leaves path on back of spoon when finger is drawn across, about 7 minutes; do not boil. Mix in chocolate, butter and corn syrup. Remove from heat; stir until smooth. Mix in sour cream. Transfer frosting to bowl and refrigerate until spreadable, stirring occasionally, about 1 hour.

Step 6

Place 1 cake layer on platter. Spread 1 cup frosting over. Repeat with second cake layer and frosting. Top with third cake layer. Spread remaining frosting over top and sides of cake, swirling decoratively.

Step 7

DO AHEAD: Can be made 1 day ahead. Cover with cake dome and chill. Let stand at room temperature 1 hour before serving.

Step 8

Sprinkle white and milk chocolate shavings thickly over top of cake.