Preheat the oven to 350°. Butter two 9-by-2-inch round cake pans and line them with parchment paper; butter and flour the paper, shaking off the excess. In a medium bowl, mix the flour, baking powder and salt. In a cup, mix the milk with the vanilla extract.
In a standing mixer fitted with the paddle, beat the butter at medium speed until light and creamy. Add the granulated sugar and beat until fluffy, about 4 minutes. Beat in the eggs, 1 at a time, scraping down the bowl between additions. Beat in the dry ingredients in 3 batches, alternating with the milk mixture.
Transfer 2 cups of the batter to a small bowl. Scrape the remaining cake batter into the prepared cake pans. Stir the melted chocolate into the reserved batter; dollop the batter over the plain batter and swirl it in with a butter knife.
Bake the cakes in the lower third of the oven for 35 to 40 minutes, until a toothpick inserted in the center comes out with a few moist crumbs attached. Let the cakes cool in the pans for 15 minutes. Run a knife around the edges and invert onto a rack. Peel off the paper, turn the cakes right side up and let cool completely.
In a standing mixer fitted with the whisk, beat the butter and egg yolk at medium-high speed until creamy. Beat in the confectioners' sugar at low speed. Beat in 6 tablespoons of the cream, the vanilla and the salt, then beat at medium-high speed until fluffy, about 3 minutes.
In a small saucepan, bring the remaining 1/4 cup of heavy cream to a boil. Remove from the heat, add the chopped chocolate and let stand until the chocolate is melted, about 3 minutes. Whisk until smooth, then transfer to a small bowl and let cool. Beat the chocolate into the frosting.
Spread 1 cup of the frosting over a cake layer and top with the second cake layer; frost the top and side. Let stand for 1 hour before serving.
The frosted cake can be stored in an airtight container at room temperature for up to 3 days.
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