1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
Makes 8-10 servings, Difficulty: Intermediate
Photograph by Kana Okada