How to Make It
Preheat the oven to 350°. Butter two 9-by-2-inch round cake pans and line with parchment paper; butter and dust with flour. In a bowl, mix the 3 cups of flour, baking powder, and salt. In a cup, mix the milk with the vanilla.
In a standing mixer fitted with the paddle, beat the butter at medium speed until light and creamy. Add the sugar and beat until fluffy, 4 minutes. Beat in the eggs, one at a time, scraping down the bowl. Beat in the dry ingredients in 3 batches, alternating with the milk mixture and scraping down the bowl.
Mix 2 cups of the batter with the melted and cooled bittersweet chocolate. Scrape the remaining batter into the pans. Dollop the chocolate batter over the batter in the pans and swirl with a knife. Bake in the lower third of the oven for about 35 minutes, until springy. Let cool for 15 minutes, then run a knife around the edges and invert onto a rack. Peel off the paper, turn the cakes upright and cool completely.
Melt the bittersweet chocolate in 1/4 cup of the heavy cream and let cool. In a standing mixer fitted with the whisk, beat the butter and egg yolk at medium-high speed until creamy. Beat in the sugar at low speed. Beat in the remainder of the cream, vanilla, and salt, then beat at medium-high speed until fluffy, 3 minutes longer. Beat in the cooled chocolate cream.
Spread 1 cup of frosting over a cake layer and top with the second layer; frost the top and sides. Let stand 1 hour; serve.