Use a large knife to carefully slice a chicken breast open lengthwise, into two pieces. Repeat with the remaining chicken breast. Once butterflied, pound the chicken breasts into 1/4-inch thick cutlets. Cover the chicken breast with a piece of plastic wrap. Use a meat mallet or rolling pin to pound it out to uniform thickness, about 1/4 inch thick. Repeat with the remaining chicken breasts. Season the chicken with S&P as desired. Set aside.
Grab 3 large plates or shallow bowls. Place the all-purpose flour on one, whisk the eggs on the second, and mix the pistachios, panko breadcrumbs, grated Parmigiano Reggiano on the third. Lightly season each plate with kosher salt and ground black pepper, mixing to combine.
Working one by one, give each chicken breast a light dusting of flour, followed by a light coating of egg, shaking off any excess after each step. Place the chicken into the pistachio-panko-Parmigiano Reggiano mixture, pressing to adhere & coat the chicken as completely as possible. Shake off any excess breading and set aside. Repeat with the remaining chicken breasts.
Add 3 Tbsp olive oil to a large skillet over medium-high heat. Once the oil is hot, give the skillet a swirl to ensure its entire surface is coated in oil. With two chicken breasts at a time, carefully add the chicken cutlets to the skillet. Cook 90 seconds - 2 minutes per side, until the pistachio coating is crispy & golden brown & the chicken is cooked through. Transfer to a plate. If needed, add the remaining 2-3 Tbsp olive oil to the skillet & repeat with the remaining two chicken breasts.