For the frosting: Place chocolate in large bowl. Heat cream, sugar, and salt in saucepan over medium-low heat, stirring occasionally, until sugar dissolves and mixture begins to simmer. Pour hot cream mixture over chocolate and whisk until smooth. Whisk in vanilla and butter until glossy. Cover and refrigerate until icing is firm but still spreadable, about 1 hour.
For the cake: Adjust oven rack to middle position and heat oven to 350 degrees. Generously grease and flour two 8- or 9-inch cake pans and line with parchment paper. Spread one-eighth of batter (about 2/3 cup) evenly in each pan and bake until edges are golden brown and cake springs back when touched, 10 to 14 minutes. Cool on rack 5 minutes. Run a knife around pan perimeter to loosen cake, then invert onto rack to finish cooling. Cool pans to room temperature, then wipe clean and repeat process 3 more times for a total of 8 layers.
To assemble: Place 1 cooled cake layer on serving platter. Spread 1/4 cup frosting over cake. (If frosting is too firm, let warm at room temperature 5 minutes, then stir to soften.) Top with second cake layer and additional 1/4 cup frosting. Repeat, alternating layers of cake and frosting and finishing with cake layer. Frost top and sides with remaining frosting. Serve.