Using clean dry beaters, beat egg whites in another large bowl until medium-firm peaks form. Gradually add remaining 2 cups sugar, beating until whites are very thick and glossy, about 5 minutes. Fold in remaining whites in 2 additions. Fold in chocolate pieces. Cool cakes in pans on racks 30 minutes.
Slide cakes onto corresponding cardboard rounds.
Stir until chocolate is almost melted. Cool chocolate to room temperature, stirring occasionally. Gradually beat in butter. Beat in cooled white chocolate in 4 additions. Beat in vanilla and almond extracts.