Preheat the oven to 350 degrees F. Butter a 9-inch-round cake pan. Fold a sheet of parchment paper into a square, then fold it in half into a triangle. Fold the triangle in half. Put the tip of the triangle in the center of the pan; trim the parchment, following the curve of the pan. Unfold the parchment circle and line the pan.
Make the cake: Sift the flour, baking powder and salt into a large bowl. Beat the eggs and sugar in a stand mixer fitted with the whisk attachment on high speed until pale and tripled in volume, about 15 minutes. Gently fold in the flour mixture in two additions using a rubber spatula, scraping down the bowl.
Add the milk, vegetable oil and vanilla and beat with the mixer until combined. Pour the batter into the prepared pan and bake until a toothpick inserted into the center comes out clean, 25 to 30 minutes. Transfer the pan to a rack and let cool 10 minutes, then invert the cake onto the rack. Invert again and cool completely.
Meanwhile, make the pudding: Heat the milk and vanilla in a medium saucepan over medium heat (do not boil). Whisk the whole eggs, egg yolks and sugar in a large bowl until light and fluffy. Add the cornstarch and salt and whisk vigorously until no lumps remain. Whisk about 1/4 cup of the hot milk mixture into the egg mixture, then gradually whisk in the remaining hot milk mixture.
Pour the egg-milk mixture into the saucepan and cook over low heat, whisking constantly, until thick and pudding-like, 10 to 15 minutes. Strain through a fine-mesh sieve into a bowl, using a rubber spatula to push the pudding through. Let cool slightly, stirring occasionally. Press plastic wrap directly onto the surface of the pudding and refrigerate at least 2 hours.
Stick toothpicks around the side of the cake as a guide for cutting two even layers, then slice the cake in half horizontally with a long serrated knife.
Place the bottom half of the cake cut-side up on a platter. Top with the pudding, spreading it to about 1/4 inch from the edge. Carefully place the other cake half on top, cut-side down, pressing gently.
Make the glaze: Heat the chocolate, cream, vanilla and salt in a saucepan over medium-low heat, stirring, until melted and smooth, about 2 minutes. Let cool about 5 minutes, then pour over the cake and smooth with an offset spatula. Refrigerate at least 30 minutes or up to 4 hours before serving.