In a medium saucepan, combine the egg yolks and 2/3 cup of sugar; whisk until well blended, then whisk in the cornstarch and 1/8 teaspoon of salt. Slowly whisk in 1 cup of milk and the cream, and add the butter.

Place the saucepan over medium-low heat and cook, stirring constantly, until the pudding thickens. Continue cooking and stirring until it comes to a gentle boil. Boil gently for about 1 minute longer. Strain through a fine mesh strainer into a bowl and add 2 teaspoons of vanilla. Stir and place a sheet of plastic wrap directly on the pudding to keep it from forming a skin. Refrigerate for 2 to 3 hours.

Heat the oven to 325 F. Butter the bottom and sides of a 9-inch round cake pan; line the pan with parchment paper and butter the paper.

Combine the cake flour, baking powder, and 1 teaspoon of salt in a bowl; whisk to blend and set aside.

Combine 3/4 cup of milk and 1/2 cup of butter in a saucepan and place it over medium heat; bring to a simmer. When the butter is melted, remove the pan from the heat and set it aside to cool slightly.

In a mixing bowl with an electric mixer, beat the eggs and sugar until thick and pale in color, about 4 to 5 minutes. The batter should fall from the beater or whisk in ribbons.

With the mixer on low, gradually add the milk and butter mixture and mix until blended.

Add the dry ingredients and 2 teaspoons of vanilla and beat on low speed just until combined.

Spoon the batter into the prepared cake pan and use a spatula to smooth the top. Bake until a toothpick inserted into the center comes out clean, about 32 to 38 minutes.

Let the cake cool in the pan on a rack for 10 minutes. If necessary, run a sharp knife around the edge of the pan to loosen the sides; carefully invert it onto the rack. Turn it over and let it cool completely on the rack.

cut the cooled cake in half horizontally using a sharp knife. For accuracy, insert several toothpicks around the side to mark the halfway point.

Put the chocolate chips, vegetable oil, and a pinch of salt in a heat-safe bowl or over simmering water—heat, stirring frequently until the chocolate has melted.

Whisk about 1/2 cup of the confectioners' sugar and 1 tablespoon of the boiling water into the melted chocolate mixture until well blended. Repeat with another 1/2 cup of confectioners' sugar and 1 tablespoon of boiling water, then repeat with the remaining confectioners' sugar and 1 tablespoon of boiling water.

Add the corn syrup and vanilla and whisk until smooth. If it is too thick, add more boiling water, a few drops at a time. If it is too thin, add more confectioners' sugar.

Spread the icing over the Boston cream pie; cover and refrigerate until serving.