Butter and flour the insides of two 9 inch (25 cm) round cake pans.
Combine the cake flour, the baking powder, and the salt, and sift them onto a piece of waxed paper or a plate. In a deep bowl cream the 6 tablespoons of butter together with the sugar, beating them until the mixture is light and fluffy.
Beat in the two eggs, one at a time, and the vanilla.
Beat in 1/3 of the flour mixture until it is thoroughly incorporated, add 1/3 of the milk, and beat until the batter is smooth.
Repeat two more times.
Divide the batter between the two prepared cake pans and bake in the center of a preheated 375F (190C) oven for about 15 minutes, or until the cake begins to shrink away from the sides of the pan and the center springs back immediately when poked with a finger.
Turn the cakes onto wire racks to cool.
For the filling, combine the cream and half the milk in a saucepan and bring almost to a boil over moderate heat.
When bubbles begin to form around the edge of the pan add the sugar and salt and stir until they are dissolved.
Remove the pan from heat.
Combine the remaining milk and the cornstarch in a bowl and stir to remove lumps.
Whisk the two eggs into this mixture, and then add the cream mixture in a thin stream, whisking constantly.
Return the contents of the bowl to the saucepan and cook over low heat, stirring constantly, until the custard thickens and is smooth.
Remove from heat; over cooking will make it lumpy.
Stir in the vanilla extract and allow to cool to room temperature.
For the frosting, stir the chocolate bits and the butter in a heavy saucepan over low heat until they are completely melted. Remove from the heat and, stirring constantly, add the cream in a thin stream.
When this mixture is smooth sift the powdered sugar over the top, and beat vigorously for a minute or two.
Stir in the vanilla extract.
To assemble the cake, spread the filling over one of the cakes with a metal spatula, and place the second cake on top.
Pour the frosting evenly over the top, allowing it to spill over the sides.