Beat the eggs and 1 tablespoon water in a shallow bowl. Set aside until ready to fry.
In a food processor, pulse the breadcrumbs with the parsley, pecorino, granulated garlic and salt until just incorporated. Transfer the breadcrumbs to a shallow dish.
Cut the chicken breasts in half crosswise. On a sturdy cutting board, place a piece of chicken between 2 pieces of plastic wrap. Using a mallet or other blunt object, pound the chicken into a flat piece about 1/8-inch to 1/4-inch thick. Repeat with the remaining chicken.
Heat 1 cup olive oil to 350 degrees F in a 14-inch cast-iron pan with the garlic cloves. Line a sheet tray with a wire rack.
Working in an assembly line, dredge the cutlets in the egg and then the breading. Double-bread by going back to the egg wash and back into the breading. Add the first piece of breaded chicken to the preheated oil along with 1/2 stick butter. Once the butter is mostly melted, add another piece of chicken. Fry until an internal temperature of 160 degrees F is reached, 2 to 3 minutes on each side. Add more oil and butter to the pan as needed while you fry the rest of your batches, allowing the oil to come back to heat between batches. Rest on the wire rack.
Meanwhile, preheat a 10-inch cast-iron pan over medium-high heat until hot, then place the lemon halves, cut-side down, directly in the pan and allow to caramelize, 1 to 2 minutes. Reserve for garnish.
Place the chicken cutlets on a large platter and garnish with freshly grated pecorino, parsley leaves and caramelized lemon halves.