Preheat your oven to 350°F (175°C) and grease a 9-inch round cake pan with butter or cooking spray, lining the bottom with parchment paper for easy removal.

Place the shelled pistachios in a food processor and pulse until they are finely ground, resembling coarse sand, but be careful not to turn them into butter.

In a medium bowl, whisk together the ground pistachios, all-purpose flour, baking powder, and salt until well combined.

In a large mixing bowl, cream the softened butter and granulated sugar together with an electric mixer on medium speed until light and fluffy, about 3 to 4 minutes.

Beat in the eggs one at a time, ensuring each is fully incorporated before adding the next, then mix in the fresh lemon zest.

Reduce the mixer speed to low and add the flour mixture in three parts, alternating with the milk, starting and ending with the dry ingredients.

Pour the thick batter into the prepared cake pan and smooth the top with a spatula to ensure it bakes evenly.

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, then transfer it to a wire rack to cool completely before glazing.

To make the glaze, whisk the powdered sugar and fresh lemon juice together in a small bowl until smooth and pourable.

Drizzle the lemon glaze over the cooled cake and let it set for 15 minutes before slicing and serving.