To Make the Pistachio Paste:
To a high-powered blender or food processor, add the pistachios. Pulse until roughly ground. Transfer to a small bowl. In a small saucepan, combine the water and sugar. Cook over high heat until it reaches 220 degrees and is bubbling. Pour over the pistachios and mix immediately.
Transfer to the blender or food processor and pulse until finely ground. While the machine is running, pour in the oil. Blend until it resembles a soft peanut butter, about 2 minutes. Transfer to a bowl, mix in the almond meal. At this point it should go from soupy to firm. Measure out 1/2 cup of the paste and store the rest in the fridge. Set aside.
To Make the Frosting:
Preheat your oven to 400 degrees. In a small baking dish, toss together the sliced plums, sugar, balsamic vinegar, salt and water. Roast for 15 minutes, or until the plums are super juicy and a syrup has formed. Set aside to cool slightly, about 5 minutes. Add the plums and its juices to a blender and pulse until completely pureed.
Pour through a strainer, into a small bowl. You should end up with about 1/3 of a cup puree. In the bowl of a stand-up mixer with the whisk attachment, add the egg whites and sugar; beat just until combined. Add a few inches of water to a saucepan and bring to a simmer. Place the stand-up mixer bowl atop the saucepan, creating a double-boiler.
Heat until the mixture reads 160 degrees on a thermometer and/or until it's very hot to the touch. Transfer the bowl back to the stand-up mixer and beat, on high, for 8 to 10 minutes until medium to stiff peaks form. The sides of the bowl should be cool to the touch. Remove the whisk attachment and in place of it, add the paddle attachment.
While the mixer is on low, add the butter, a tablespoon at a time, until it's light and smooth and fluffy. Lastly, add the 1/3 cup of plum puree and mix until combined. If the frosting is soupy (mine was because it was really hot), add the bowl to the fridge to chill for 5 to 10 minutes. This will work like a charm. Set aside until you're ready to frost the cake.
To Make the Italian Pistachio Cake Layers:
Preheat oven to 350 degrees F. Butter and flour two 8x3-inch cake pans. Line the bottoms with a round of parchment (this isn't totally necessary but I always do this because my biggest fear is cake sticking to the bottom of pans). Set aside.
In a medium bowl, whisk together the flour, baking powder and salt. In the bowl of a stand-up mixer add the butter and sugar; cream together until light and fluffy, about 2 minutes. Add the zest and then crack in one egg at a time, adding the next egg only when the one before it has combined.
Almost lastly, add in the vanilla extract. Turn the mixer down to low speed and add the yogurt and flour mixture in a few batches, alternating between the two. Lastly, add the pistachio paste and mix until properly combined.
Divide the cake batter amongst pans, smoothing out the top with a spatula (the batter will be a little thick so it'll need some help smoothing out). Transfer to the oven and bake for 40 to 45 minutes, or until a a skewer inserted into the center comes out clean. Cool the cakes in their pan for 10 minutes before removing them to cool completely on a wire rack.
To Assemble the Cake:
Place the first layer of cake on a cake board or a cake stand. Add about 1/4 cup of frosting the top of the cake layer and smooth it out, pushing it out to the sides. Add the second layer on top and then add a second 1/4 cup to the top of that. (These measurements can be eyeballed.)
Smooth out the top and add a nice thin layer all the way around the sides of the cake. Transfer to the freezer to chill for about 5 to 7 minutes. This is the crumb coat. Add a second even layer on the outside. Transfer to the cake to a baking sheet and press the chopped pistachios all around the sides of the cake. Slice and serve!