To make the Cake: Heat oven to 350 degrees. Line the bottom of an 8-inch round cake pan with parchment paper. Butter and flour the sides of the pan.
Beat the butter and sugar together at medium-high speed until fluffy. Beat in the eggs and yolks one at a time. Add the vanilla extract.
Stir the cake flour, baking powder, and salt together using a whisk. Reduce mixer speed to low and beat the flour mixture into the butter-egg mixture, adding it in thirds and alternating with the milk. Beat until the batter is smooth.
Transfer to the prepared pan and bake on the center shelf of the oven until the cake tests clean when a skewer is inserted into the center, 35-40 minutes. Cool completely on a wire rack.
Make the Pudding while the cake is cooking: Heat the milk, cream and vanilla bean (with the seeds scraped into the milk) until just simmering in a heavy bottomed medium pot.
Whisk the yolks, cornstarch, sugar and salt together and stream the hot milk mixture, while whisking continuously, into the eggs. Return to the pan and cook, while whisking over medium heat until the mixture thickens and just begins to boil. Immediately transfer to a clean bowl. Pick out and discard the vanilla bean. Stir in the vanilla and the butter. Press plastic wrap against the surface to cover. Cool completely before assembling cake.
To make the glaze: When you are ready to assemble the cake make the glaze. Heat the heavy cream to a boil. Place the chocolate, butter and corn syrup in a medium size heatproof bowl and add he hot milk to it. Let sit for 1 minute and stir until smooth. Use while still warm to assemble the cake.
To Assemble: Splite the unmolded cake, to make two layers, using a serrated knife. Spread the pudding over the bottom half of the cake and place the top layer over the pudding. Pour Chocolate Glaze over the cake, allowing it to drip down the sides of the cake.