Butter two 6-inch round cake pans and line the bottoms with parchment. Lightly dust sides with flour. Preheat oven to 350°F (175°C), placing a rack in the center. This ensures even baking and easy release for small, delicate layers.

In a food processor, pulse 2/3 cup pistachios until very finely ground but not pasty. In a medium bowl, whisk the ground nuts with cake flour, baking powder, baking soda, and salt. Set aside; combining these ensures nuts distribute evenly and leavening activates properly.

In a stand mixer with paddle attachment, cream butter, oil, and sugar on medium speed until pale and fluffy, 2-3 minutes. The combination of butter and oil gives richness and keeps the crumb tender without greasiness. Scrape down the bowl as needed.

Add egg yolks, vanilla bean paste, and almond extract. Beat on medium-low until fully combined and creamy, about 45 seconds. Room temperature yolks blend smoothly and help emulsify fats for even crumb and flavor.

With mixer on low, add flour-nut mixture in three stages, alternating with buttermilk, beginning and ending with flour. Mix each addition just until combined. Avoid overmixing to preserve cake tenderness; finish folding by hand if needed for even texture.

In a clean bowl, whip egg whites to soft peaks. Gently fold 1/3 of the whites into batter to lighten, then fold in remaining whites just until no streaks remain. This step adds lift and an airy structure to the cake layers.

Divide batter evenly between prepared pans, smoothing tops with an offset spatula. Tap pans to release bubbles. Bake 24-28 minutes, rotating after 15 minutes, until tops spring back and a toothpick emerges with moist crumbs. Do not overbake; cakes should look lightly golden.

Transfer pans to a wire rack and cool 15 minutes. Carefully invert cakes out of pans; peel off parchment. Let layers cool fully on racks before filling or frosting, warm cake will cause the frosting to melt and slip.

Cream butter and cream cheese together until smooth. Add powdered sugar, ground pistachios, salt, and 2 tsp milk. Beat until fluffy, adding more milk a few drops at a time as needed for desired spreadable texture.

If needed, level cooled cake layers with a serrated knife. Place first layer on a serving plate. Pipe or spread 1/3 of the frosting evenly over, then top with the second layer. Smooth remaining frosting over top and sides. Decorate with reserved chopped pistachios.