In a high-powered blender or food processor, blend pistachios. Keep blending and scraping down the sides until a paste forms. With blender running, drizzle in olive oil. Keep blending till a smooth, looser paste/butter forms. Set aside to use later.
Preheat oven to 350 degrees F. Butter and line with parchment 3 or 2 8 inch cake pans.
In a stand mixer fitted with paddle attachment, cream butter for 3 minutes. Add sugar and beat for 3 more minutes. Add pistachio paste and vanilla and beat for another 30 seconds.
Sift together cornstarch and flour. Add baking powder and salt.
Add half of this flour mixture and half of the buttermilk. Mix for 15 seconds. Add the other half of the flour and the other half of the milk, mix for another 15 seconds.
In another stand mixer bowl fitted with whisk attachment, beat egg whites until soft peaks form. Slowly add sugar as you keep whipping the egg whites. Whip until stiff peaks form.
Slowly and carefully fold the egg whites into the flour, butter, and milk mixture until barely any visible egg white remains. Pour into the prepared cake pans and bake for 25 to 30 minutes.
In stand mixer bowl fitted with paddle attachment beat together butter and pistachio paste. Add powdered sugar and keep beating. Add 1 tablespoon of milk and continue to mix. Add another tablespoon if mixture seems too thick.
Place a dab of frosting on a cake plate or cake round and then place the first later of cake down. Use a generous smattering of frosting to top the cake round and then place the next layer on top, repeating topping with frosting. Frost top and sides of cake.