Step 1

Position rack in center of oven; preheat to 350°F. Butter two 9-inch-diameter cake pans with 2-inch-high sides. Dust pans with flour; tap out excess. Whisk first 4 ingredients in medium bowl to blend. Using electric mixer, beat sugar and butter in large bowl until well blended. Beat in eggs 1 at a time, beating well after each addition. Beat in yolk. Add cocoa and beat until well blended. Add flour mixture in 3 additions alternately with ice water in 2 additions, beginning and ending with flour mixture and beating until just blended and smooth after each addition. Divide batter between prepared pans; smooth tops.

Step 2

Bake cakes until tester inserted into center comes out clean, about 40 minutes. Cool cakes in pans on racks 15 minutes. Invert cakes onto racks and cool completely. DO AHEAD: Can be made 1 day ahead. Wrap in foil; store at room temperature.

Step 3

Bring cream and corn syrup to simmer in medium saucepan. Remove from heat; add chocolate and whisk until melted and smooth. Transfer to small bowl. Chill until firm enough to spread, about 1 hour. DO AHEAD: Can be made 1 day ahead. Before using, let stand at room temperature until soft enough to spread, about 30 minutes.

Step 4

Place white chocolate in large heatproof bowl. Bring 1 cup cream to simmer in saucepan. Pour hot cream over white chocolate. Let stand 1 minute; whisk until smooth. Whisk in extract. Cover; chill until mixture thickens and is cold, at least 4 hours. DO AHEAD: Can be made 1 day ahead. Chill.

Step 5

Add 2 cups chilled cream to white chocolate cream and beat until smooth and peaks form. DO AHEAD: Can be made 3 hours ahead. Cover and chill. Rewhisk to thicken, if necessary, before using.

Step 6

Using long serrated knife, cut each cake horizontally in half. Place 1 cake layer on platter, cut side up. Spread 1/3 of dark chocolate ganache over cake. Spoon 2 cups white chocolate cream in dollops over cake; spread evenly to edges. Top with second cake layer, cut side down; spread 1/3 of ganache over, then 2 cups white chocolate cream. Repeat with third cake layer, cut side up, remaining ganache, and remaining cream. Cover with fourth cake layer, cut side down. Chill while preparing frosting.

Step 7

Combine sugar, 1/2 cup water, egg whites, and corn syrup in large bowl of heavy-duty stand mixer. Whisk by hand to blend well. Set bowl with mixture over saucepan of gently simmering water; whisk constantly with hand whisk until mixture resembles marshmallow creme and ribbons form when whisk is lifted, 8 to 9 minutes. Whisk in peppermint extract. Remove bowl from over water and attach bowl to heavy-duty stand mixer fitted with whisk attachment. Beat on high speed until mixture is barely warm to touch and very thick, 7 to 8 minutes.

Step 8

Using offset spatula and working quickly, spread frosting over top and sides of cake. Sprinkle chocolate curls over top and sides. DO AHEAD: Can be made 1 day ahead. Cover with cake dome; chill.