Preheat your oven to 350°F (175°C). Lightly grease an 8-inch round cake pan and line the bottom with parchment paper to ensure the cake releases easily.

Place the shelled pistachios in a food processor and pulse until finely ground, resembling coarse sand. Be careful to pulse in short bursts so the nuts do not turn into pistachio butter.

In a medium bowl, whisk together the ground pistachios, all-purpose flour, baking powder, and salt until evenly distributed. Set this dry mixture aside.

In a large mixing bowl, beat the eggs and granulated sugar together using a hand mixer on medium speed for about 3 minutes, until the mixture is visibly pale and fluffy.

Gradually pour the extra virgin olive oil into the egg mixture while continuing to beat on low speed. Add the whole milk, lemon zest, and vanilla extract, mixing just until combined.

Gently fold the dry pistachio and flour mixture into the wet ingredients using a rubber spatula. Stir just until no dry streaks of flour remain to ensure a tender, moist cake.

Pour the batter into the prepared cake pan, smoothing the top with your spatula. Bake for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean and the edges are golden brown.

Allow the cake to cool in the pan for 10 minutes before gently turning it out onto a wire rack to cool completely. Dust generously with powdered sugar before slicing, and serve with a hot cup of coffee or tea.