Preheat an oven to 450 F.
Debone the breast off the bird, keeping the breast whole. Remove the legs from the body and discard the skin. Cut the leg in half, saving the drumstick for another use. Remove the thighbone from the meat and discard the bone.
Put the boneless thigh meat and innards through a sausage grinder. Place the ground chicken mixture into a medium mixing bowl, add the egg, cheese and pistachios and mix well.
Take the breast, sandwich between two sheets of plastic wrap and pound it to 1/4-inch thick. Season the chicken breast with salt and place a prosciutto slice over each side of the breast. Form the stuffing into a log shape and place over the widest part of the breast, so it will fit like a log when the breast is rolled up. Tightly roll up the breast around the stuffing and roll the skin around the breast tightly. Secure the bundle by tying butchers twine around the rolled meat at intervals of one and a half inches.
In a medium sauté pan over medium high heat, melt the butter and warm the olive oil. Place the chicken in the pan. Sear the outside of the breasts until golden brown, 2 to 3 minutes per side. Place the pan in the oven and bake until the chicken is cooked through and an instant read thermometer registers 160 F, 12 to 15 minutes.
Let the meat rest for at least 5 minutes before cutting into thin slices and serving.
Adapted from Chef Marc Vetri, Vetri Ristorante, Osteria, Philadelphia