Pat the 1 ½ lb boneless skinless chicken breasts dry and cut them into strips about ¼ inch thick.
Season both sides of the chicken strips with salt and pepper.
Put ½ cup all-purpose flour on a large plate for dredging.
Beat 2 eggs in a shallow bowl.
On another large plate, mix ⅔ cup breadcrumbs with ½ cup finely grated Parmigiano-Reggiano.
Working with one piece at a time, dredge each chicken strip in the flour and shake off excess, dip it into the beaten eggs, then press it into the breadcrumb–Parmesan mixture to coat evenly. Repeat until all pieces are coated.
Heat a skillet over medium heat and add enough vegetable oil to coat the bottom of the pan. When the oil is hot, fry the chicken in a single layer without crowding the pan, 2–3 minutes per side, until golden brown and cooked through. (Cook in batches if needed.)
Transfer the cooked chicken to a plate lined with paper towels to drain briefly, then serve.