Preheat oven to 350 degrees. Butter two 9-inch cake pans and line with parchment. Butter the parchment and dust the pans with flour, tapping out excess.
Pulse pistachios in a food processor until coarsely ground. Add flours and salt and pulse until finely ground.
Combine the yolks, 1 cup of sugar, and vanilla in a large bowl and whisk until pale and fluffy, about two minutes. Whisk in the pistachio paste and melted butter.
Using a mixer, whisk the whites on medium speed until soft peaks form. Raise speed to medium-high and gradually add remaining 1/2 cup of sugar. Whisk until stiff, glossy peaks form, about 3 to 4 minutes.
Carefully fold the egg whites into the egg yolk mixture. Then, working in three additions, fold in the flour mixture. Transfer the batter to the prepared pans and bake until cakes are set and lightly golden, about 25 minutes.
Let cool 10 minutes in pan. Run a knife around the pans to loosen the cakes and invert onto a wire rack. Remove parchment and reinvert to cool completely.