Position a rack in the lower third of the oven and preheat to 350 degrees ;. Grease two 8-inch round cake pans. Line the bottoms of the pans with parchment paper, then grease the parchment. In a medium bowl, whisk together the flour, baking powder, baking soda and 1/2 teaspoon salt.
In a saucepan, dissolve 2 cups sugar in 2 cups water over medium-high heat. Remove from the heat, add the coarsely chopped chocolate and 6 tablespoons butter and whisk until smooth. Whisk in the cocoa powder. Beat in the eggs, then stir in 2 teaspoons vanilla. Add the flour mixture, whisking until incorporated. Divide the batter between the prepared pans and bake until the cakes are springy to the touch and have pulled away from the sides of the pans, about 30 minutes. Transfer to racks and let cool for 10 minutes. Remove from the pans and place on the racks. Let cool completely.
Meanwhile, in a bowl, combine the finely chopped chocolate and the remaining 9 tablespoons butter, 2 teaspoons vanilla and 1/8 teaspoon salt. In a medium saucepan, bring the cream and remaining 1 1/2 cups sugar to a boil. Pour the cream mixture over the chocolate and let stand for 2 minutes, then stir the mixture until smooth. Refrigerate, stirring occasionally, until thickened to a spreadable consistency, about 1 hour 15 minutes.
Lay 4 strips of parchment paper around the border of a cake stand. Remove the parchment from 1 of the cake layers and place the layer on the cake stand. Place about 1 cup of the chocolate frosting in the center of the layer and spread to within 1/4 inch of the edge. Remove the parchment from the second cake layer and place the layer on top of the first. Apply a thin coating of frosting around the sides of the cake, rotating as you go. Top the cake with the remaining frosting and spread from the top of the cake down and around the sides. Discard the parchment paper strips.
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