Preheat the oven to 350° and position racks in the lower and middle thirds. Spray two 12-by-17-inch baking pans with vegetable oil spray and line the bottoms with parchment paper. Spray the parchment.
Put the 8 egg whites in the bowl of a standing electric mixer fitted with the whisk. Add 2 cups of the confectioners' sugar, the vanilla seeds and salt and beat at medium-high speed until the whites are stiff and glossy, 5 minutes. Scrape the meringue into a medium bowl.
Clean the mixing bowl and add the 11 egg yolks along with the remaining 2 cups of confectioners' sugar, the flour and the milk. Beat at medium speed until smooth, about 3 minutes. Beat in one-fourth of the beaten eggs whites to lighten the batter. Using a rubber spatula, fold in the remaining egg whites until no streaks remain.
Spread the batter evenly in the prepared baking pans (about 5 cups for each). Bake for 12 to 15 minutes, until the cakes are golden and set; shift the pans from top to bottom and front to back halfway through baking. Transfer to racks to cool.
Put the chopped chocolate in a medium bowl. In another medium bowl, whisk the 3 egg yolks with the sugar and salt until pale, 2 minutes. In a medium saucepan, heat the half-and-half with the cocoa powder until hot. Whisk the hot liquid into the yolks, then scrape the mixture back into the saucepan. Cook over moderate heat, whisking constantly, until slightly thickened and an instant read thermometer registers 160°, 4 minutes. Strain the mixture over the chopped chocolate and let stand until melted. Whisk until smooth, then let cool completely.
In a mixing bowl fitted with the whisk, beat the butter until creamy. Scrape the chocolate mixture into the butter and beat until smooth and creamy.
Lightly dust the cakes with confectioners' sugar and top with a sheet of wax paper. Invert the cakes onto a work surface and carefully peel off the parchment. Cut each layer crosswise into four 12-by-4 1/4-inch strips. Spread 6 layers with 1/2 cup of buttercream each and stack them on a rectangular cake cardboard or cake plate. Top with a seventh layer (save the eighth layer for a snack). Frost the top and sides with the remaining buttercream and refrigerate the cake until well chilled, at least 2 hours or preferably overnight.
In a small saucepan, combine the sugar with 1/2 cup of water and bring to a boil, washing down the side of the pan with a moistened pastry brush. Boil the syrup until it registers 240° on an instant-read thermometer. Meanwhile, in a clean standing electric mixer bowl, beat the egg whites with the salt until soft peaks form. Beating at medium speed, slowly drizzle in the hot syrup and beat until incorporated, then beat at high speed until the meringue is stiff and glossy and the side of the bowl is no longer warm, about 7 minutes.
Spread a thin layer of the meringue all over the cake, then use the remaining meringue to form decorative swirls on the cake. Using a brulée torch, brown the meringue all over. Refrigerate the cake for at least 1 hour before serving.