Preheat the oven to 350 F. Grease a 13x18 inch sheet pan or 9x13 baking dish with nonstick spray or butter. If using a sheet pan, measure and cut a 13”x 26” inch sheet of parchment paper and place into the pan so that 4 inches of overhang remain on the sides of the pan. If using a 9x13 dish, cut a 9”x21” sheet of parchment paper and follow the same placement, leaving an overhang on the sides of the dish.
Add the softened butter, oil, and sugar to the bowl of a stand mixer fitted with a whisk attachment. Whip fats and sugars together for 3-4 minutes, until pale and sugar has mostly absorbed.
Add one full egg and vanilla extract to the stand mixer bowl. In a separate bowl, separate four egg whites from the egg yolks. Add the egg whites to the rest of the ingredients and set the egg yolks aside in the fridge for making the custard later. Whip the eggs and egg whites into the creamed butter for another 4-5 minutes, until fluffy and white.
Place the pistachios in a blender or food processor and pulse for 2 minutes, until they are mostly crumby and sandy, but a few larger grains may remain. Set aside ¼ cup of the ground pistachios for decorating later. Place the ground pistachios in a large mixing bowl. Add the flour, baking powder, baking soda, and salt to the pistachios and whisk everything together until cohesively mixed. Measure 1 cup of buttermilk and set aside in a small bowl or glass measuring cup.
Combine all the batter ingredients by alternating additions of the dry mix and the buttermilk into the stand mixer bowl: Remove the bowl from the stand mixer and use a rubber spatula to hand-stir and scrape in between each addition. Add ⅓ of the dry ingredients to the bowl and stir until mostly combined. Scrape the bottom of the bowl and fold everything together. Add ½ of the buttermilk and repeat the stirring and scraping. Add another ⅓ of the dry ingredients and stir and scrape the bowl. Repeat additions of and stirring with the remaining buttermilk and dry ingredients. Once no streaks of liquid or dry remain, Pour the batter into your greased and lined pan.
Bake for 30-35 minutes, or until the center top of the cake is deep golden brown and bouncy to touch. If the cake is browning but not yet set in the middle, place a large sheet of aluminum foil lightly on top of the cake while it finishes baking.
Place the sugar, cornstarch, and salt into a medium sized saucepan and whisk together until the sugar granules have broken up any clumps of cornstarch. Set aside
Whisk the 4 remaining egg yolks from the cake recipe with 2 cups of milk in a medium bowl until no streaks of yellow remain. Measure the butter and vanilla and set aside in a small bowl for later.
Place the saucepan on medium heat and slowly pour in the egg and milk mixture. Whisk constantly while the custard comes to a simmer and it begins to foam. Keep whisking, scraping the bottom of the pan, while the custard begins to bubble, after a total of 5-7 minutes. Take the pan of the heat when the custard thickens to a brownie batter consistency and the whisk leaves visible traces in the custard. Add the butter and vanilla extract and continue to whisk as it finally thickens to a pudding-like thickness. If the custard is not holding onto the whisk, keep it on low heat for another 1-2 minutes until properly thickened.
Immediately transfer the custard into an 8x8” baking dish or large, wide bowl. Cut a large sheet of saran wrap and press it onto the surface of the custard, making sure every inch of exposed custard is making contact with the saran wrap. This prevents the custard from forming a “skin” while it cools in the fridge for 2 hours, or until cold to touch. The custard can be made and stored in the fridge up to 3 days in advance
Once the cake and custard are ready to assemble and serve, remove the cake from the pan using the overhang of parchment paper to lift the cake out and onto a large board. Alternatively, the cake could be served right in its baking vessel, as well!
In a large bowl, whip the heavy cream for 3-5 minutes, or until stiff peaks form. Remove the custard from the fridge and use a spatula to stir it and loosen it up. Add ¼ of the whipped cream to the custard and stir it together to further lighten the custard. Once stirred, add the lightened custard into the whipped cream and gently fold together.
In a small bowl, stir the jam, lemon juice, and salt together until the jam is runnier and less sweet to taste.
Assemble the cake by slathering on the custard cream and dolloping spoons of the jam mix. Swirl with a knife or spoon to achieve red swoops and swirls in the cream. Decorate with fresh raspberries and reserved ground pistachios. Slice and serve immediately! Cake can be wrapped and stored in the fridge for up to 5 days.