Preheat the oven to 350°F (180°C). Line the bottoms of two 9-by-2-inch (23-by-5cm) round cake pans with parchment (baking) paper.
Sift the flour, baking powder, and salt together onto a sheet of waxed paper; set aside. In a small bowl, combine the milk and vanilla; set aside.
Using a stand mixer, beat the butter with the paddle on medium speed until creamy. Add sugar and beat until mixture is pale and fluffy. Drizzle in the eggs beating each addition until incorporated before continuing (Stop frequently to let the batter absorb the egg. If the batter begins to look curdled , increase the mixer speed to high and beat until batter looks smooth again.). Reduce the speed to medium low and add the dry ingredients in 3 additions alternately with the milk mixture in 2 additions, starting and ending with dry ingredients. Beat just until combined.
Pour the batter into prepared pans and spread it evenly. Bake until the cakes are puffed and a skewer inserted into the centers comes out clean, 20-25 minutes. Let cool completely on a wire rack. Run a table knife around the edges of the pans and invert the cakes unto plates.
To make the frosting, melt the chocolate and let cool to room temperature (To melt chocolate, chop it into small pieces and put it in a stainless steel bowl. Set the bowl in a saucepan over, but not touching, barely simmering water. Heat until the chocolate melts, stirring occasionally. Do not allow any water or steam to come in contact with the chocolate, or it will become stiff and grainy. Or, chop the chocolate in a large pieces, place in a microwave-safe dish, and microwave on low for 1 minute. Continue to microwave if necessary, checking every 20 seconds, until the chocolate looks softened, then stir until smooth and liquid.).
Using a stand mixer, beat the butter with the paddle on medium speed until it is the consistency of mayonnaise. It should not be melted. Transfer to another bowl and thoroughly wash and dry the mixer bowl. In a saucepan over medium heat, heat the milk and 1/4 cup (2 oz./60 g) of the sugar, stirring occasionally, until small bubbles appear along the edge of the pan.
Meanwhile, using the stand mixer, beat the egg yolks and the remaining 1/2 cup (4 oz./100 g) sugar with the whisk on medium-high speed until the mixture is pale and thick, about 3 minutes. Reduce the speed to low and pour in the hot milk mixture in a thin stream. Return the mixture to the saucepan. Thoroughly wash and dry the mixer bowl and whisk.
Cook over medium heat, whisking constantly until it registers 170°F (77°C) on an instant read thermometer 5-7 minutes. Pour the mixture back into the mixer bowl and heat with the whisk on medium speed until cool, 5-10 minutes. Beat in the melted chocolate. Add the butter in 4 additions, incorporating each addition before adding another.
To frost the cake, put one layer, top side down, on a serving plate (To protect the plate, place paper strips under and along the edges of the bottom layer.). Peel off the parchment paper. Using a straight frosting spatula, spread evenly with about one third of the buttercream. Invert the other layer, top side down, on the first layer peel off the paper. Refrigerate the cake 30 minutes to firm the frosting; keep the remaining buttercream at room temperature. Spread the buttercream on the top and sides cake (Put the frosting in the center of the top of the cake and smooth it evenly. The frosting will cover the top in a thick layer. Evenly smooth the frosting from the top of the cake down the sides, turning the cake as you work. As you cover the side, hold the spatula almost perpendicular to the top of the cake. Holding the spatula almost horizontal to the cake's top, smooth the top with long strokes. Then run the spatula around the sides, making the frosting as smooth as possible. Using short strokes, remove excess frosting from the edge of the top. Discard the waxed paper strips.). Refrigerate the cake until 30 minutes before serving to set the frosting.