Prep: Preheat oven to 350°F. Line three 6" cake pans with rounds of parchment paper on the bottom and set aside.
Grind pistachios: In a bullet blender or food processor, whiz pistachios for about 30 seconds until they become a coarse flour texture. Transfer this to a medium bowl.
Dry mixture: To the pistachios, add flour, baking powder, baking soda, and salt. Stir together to combine. Set aside.
Wet mixture: In a large mixing bowl, add melted butter and sugar. Whisk together until combined. Add the eggs, one at a time, whisking until incorporated.
Wet mixture continued: Add the greek yogurt, oil, and vanilla extract and whisk again until combined.
Combine dry and wet mixture: Use a fine-mesh sieve to sift the flour mixture into the wet mixture. The goal is just to break up any clumps of pistachio flour. Any larger pieces of pistachios that are remaining in the sieve can be added back to the batter. Whisk together until just combined.
Fill cake pans: Divide the batter into your three lined cake pans (about 320g of batter in each). Give each of the pans a spin on the counter to push the batter to the sides of the pan (a trick to help the cakes bake flatter).
Bake: Bake cakes for about 30-35 minutes or until the tops are lightly golden brown and a toothpick inserted in the middle comes out mostly clean.
Release cake layers: Let the cakes cool in the pans until they're cool enough to touch with your hands. Run an offset spatula around the sides of each cake and invert the pan to release the cake layers onto a wire rack. Peel the parchment paper off and let the cakes cool completely before assembling.