Preheat oven to 350°F Butter two 9-inch-diameter cake pans with 1 1/2-inch-high sides. Line bottoms with parchment paper; butter parchment. Stir unsweetened chocolate in heavy medium saucepan over low heat until melted and smooth. Cool slightly. Sift flour, cocoa powder, baking powder, baking soda, and salt into medium bowl. Using electric mixer, beat sugar and butter in large bowl until well blended. Beat in lukewarm melted unsweetened chocolate and vanilla. Beat in eggs, 1 at a time, blending well after each addition. Mix in cocoa powder mixture in 3 additions, alternating with milk in 2 additions.
Divide batter between prepared pans. Bake cakes until tester inserted into center comes out clean, about 35 minutes. Cool cakes in pans on racks 10 minutes. Run knife around edge of each pan to loosen cakes. Turn cakes out onto racks and cool completely. Peel off parchment.
For filling and frosting:
Bring 1 cup cream and butter to simmer in heavy medium saucepan over medium heat, stirring until butter melts. Remove from heat. Add chocolate and stir until smooth. Whisk in vanilla. Pour 1 1/2 cups chocolate filling into small bowl; cover and freeze until cold, about 2 hours.
Let remaining chocolate mixture in saucepan stand at room temperature until lukewarm, about 20 minutes.
Using electric mixer, beat remaining 2 cups cream and almond extract in large bowl until peaks form. Working in 3 batches, fold in lukewarm white chocolate mixture. Refrigerate whipped cream frosting just until medium peaks hold, folding occasionally, about 3 hours.
Place 1 layer on platter. Spread chilled 1 1/2 cups chocolate filling over. Top with second cake layer. Spread whipped cream frosting over top and sides of cake. Chill until cold and set, about 1 hour. (Can be made 1 day ahead. Cover and keep refrigerated.).