Pat the 1 ½ lb. chicken breasts dry with paper towels, slice each breast into strips or cutlets about 1/4-inch thick, and season both sides with salt and pepper.
Place ½ cup all-purpose flour in a shallow dish for dredging. Beat the 2 eggs in a second shallow dish. In a third shallow dish, combine ⅔ cup breadcrumbs and ½ cup finely grated Parmigiano-Reggiano cheese.
Working with one piece at a time, lightly coat the chicken in the flour, shaking off any excess.
Dip the floured chicken into the beaten eggs, allowing excess egg to drip off.
Press the egg-coated chicken into the breadcrumb–Parmesan mixture so it is evenly coated on all sides.
Heat a thin layer of vegetable oil in a large skillet over medium heat until shimmering. Fry the chicken in a single layer without crowding the pan, working in batches if needed.
Cook the chicken 2–3 minutes per side, until the coating is golden and the chicken is cooked through (no pink inside or an instant-read thermometer registers 165°F).
Transfer cooked pieces to a paper towel–lined plate or wire rack to drain and let rest for a couple of minutes before serving.