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.
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.
For dark chocolate ganache:.
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.
For white chocolate cream:
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.
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.
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.
For peppermint frosting:
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.
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.