Preparation Steps

• Position a rack in the middle of the oven and preheat to 350°F (175°C). Line a 9x13-inch pan with parchment (leave an overhang) and lightly spray the paper and sides.
• In the bowl of a stand mixer fitted with the paddle (or using a large bowl and electric hand mixer), beat the butter and 1 1/3 cups sugar on medium-high until pale and fluffy, about 3–4 minutes. Scrape the bowl.
• With the mixer running on low, stream in the oil, then add the eggs one at a time, beating briefly after each so the mixture stays emulsified. Mix in the yogurt and 1 teaspoon vanilla until smooth.
• Whisk together the flour, baking powder, baking soda, and salt in a separate bowl. Sift or add in two additions to the mixer and fold gently with a spatula until just incorporated—avoid overmixing to keep the cake tender.
• Scrape the batter into the prepared pan, smooth the top, and bake 30–36 minutes (I used 34 minutes) until the center springs back and a toothpick comes out with a few moist crumbs. Do not overbake.
• While the cake bakes, prepare the pastry cream: whisk the 8 egg yolks and 1 cup sugar in a bowl until bright and slightly thickened. Whisk in the cornstarch until smooth.
• Heat the 4 cups of milk with the lemon zest in a saucepan just to a simmer. Slowly pour roughly a cup of the hot milk into the yolk mixture while whisking constantly to temper the yolks, then pour the tempered mixture back into the saucepan.
• Return the pan to medium heat and whisk continuously until the custard boils and thickens to a pudding-like consistency (about 1 minute after it first boils). Remove from heat, stir in the cubed butter and vanilla until smooth. Strain through a fine mesh into a bowl.
• Cover the surface tightly with plastic wrap to prevent a skin and plunge the bowl into an ice bath to bring it down to near-room temperature quickly. Once cooled, divide the pastry cream: reserve half for filling the cake holes and chill the other half for frosting.
• As soon as the cake comes out of the oven and is still warm (about 8–12 minutes after removing), use the handle of a wooden spoon to poke holes across the surface, spacing them an inch or so apart. Pour the reserved warm pastry cream evenly over the cake, pressing gently so the cream sinks into the holes.
• Allow the cake to cool to room temperature (about 30 minutes) so the filling sets into the cake while you finish the frosting and ganache.
• Make the diplomat frosting: whip the cold 1/2 cup heavy cream to soft peaks. Lightly loosen the chilled half of the pastry cream with a whisk and fold in the whipped cream gently until smooth and airy. Spread this frosting over the cooled cake and smooth the top.
• For the ganache, place the chopped chocolate in a bowl. Heat 1 cup heavy cream until it just begins to simmer, then pour over the chocolate. Let sit 1 minute, then whisk until glossy and smooth. Stir in 1 tablespoon neutral oil for extra sheen and let the ganache cool until slightly thickened but still pourable, about 10–15 minutes.
• Pour the ganache over the frosted cake and spread to the edges with an offset spatula. Chill the finished cake in the refrigerator at least 3 hours (or overnight) to firm up before slicing into 12 squares.
• Slice and serve chilled or slightly cool-from-fridge. Store leftovers covered in the refrigerator for up to 2 days.