1. Toast the pistachios. Preheat the oven to 350°F. Spread the pistachios in an even layer on a rimmed baking sheet. Toast in the preheated oven for 10 minutes. Let cool slightly.
2. Prepare the pan. While the pistachios are toasting, brush a high-sided 8-inch round cake pan or springform pan with olive oil. Line the bottom of the pan with parchment paper and brush the parchment with a thin layer of olive oil and set aside.
3. Grind the pistachios. Transfer the pistachios to a food processor. Add 1 tablespoon of the sugar to the pistachios. Process until finely ground, about 1 to 2 minutes.
4. Mix the dry ingredients. In a medium mixing bowl, whisk together the ground pistachios, flour, baking powder, baking soda, and salt.
5. Mix the wet ingredients. In a large mixing bowl, whisk together the Greek yogurt, the remaining sugar, eggs, olive oil, and vanilla extract until thoroughly combined.
6. Make the batter. Add the flour mixture to the yogurt mixture and use the whisk to stir gently just until no streaks of dry flour remain.
7. Bake the cake. Pour the batter into the prepared pan. Bake for 30 to 32 minutes, or until a toothpick inserted into the center of the cake comes out clean.
8. Cool the cake. Let the cake cool in the pan for 15 minutes. Then, turn it out onto a wire rack to cool completely, about 1 hour.
9. Make the Pistachio Butter Glaze. In a small bowl, whisk together the pistachio butter, powdered sugar, and water until smooth. The glaze should be thick but pourable. Add extra water, a teaspoon at a time, to thin to your preferred consistency.
10. Glaze the cake. When the cake has cooled completely, transfer it to a cake stand or serving plate. Pour the glaze on top of the cake and use an offset spatula or the back of a spoon to coax the glaze to the edges. As it sets, the glaze will slowly drip down the sides of the cake, and the top of the glaze will level and become smooth. Sprinkle the top with chopped pistachios. For clean slices, let the glaze set for at least 30 minutes before slicing and serving.