Preheat oven to 350°F and grease a 9-inch springform pan with olive oil. Line the bottom with parchment for easy release and even baking.
In a bowl, whisk together flour, baking powder, sugar, ground pistachios, lemon zest, and salt until fragrant and evenly combined.
In a separate bowl, whisk eggs until pale, then slowly pour in olive oil, whisking until smooth and glossy to create a silky batter base.
Gently fold the wet egg mixture into the dry ingredients using a spatula until just combined, being careful not to overmix.
Pour the batter into your prepared pan, smooth the top with the spatula, then gently tap the pan on the counter to release any large air bubbles.
Bake for 30–35 minutes until the top is golden and a toothpick inserted in the center comes out clean.
Allow the cake to cool in its pan on a wire rack for 10 minutes, then gently remove the springform ring and cool completely for at least 1 hour.
Whisk together powdered sugar and fresh lemon juice until smooth and pourable, adjusting consistency with more sugar or juice.
Drizzle the lemon glaze over the completely cooled cake, allowing it to drip down the sides.