Preheat oven to 350°F. Prepare a cupcake pan with cupcake liners.
Place ½ cup pistachios in a food processor. Pulse until the texture resembles small, coarse gravel. Transfer to a small bow and set aside. Place the remaining ½ cup pistachios in the food processor. Pulse until the texture resembles coarse sand.
In the bowl of an electric mixer, add finely ground pistachios, flour, baking powder, salt and sugar. Mix ingredients on the lowest setting until combined. Add extra virgin olive oil and whole milk and blend on a medium setting. Mix in eggs and lemon juice. Finally, add the coarsely ground pistachios and stir on a low setting.
Pour batter into the cupcake liners. Bake for about 25 – 30 minutes. You will know your cupcakes are done when an inserted toothpick comes out clean. Let cool.