Slice each chicken breast in half widthwise to create two thinner cutlets (or get three cutlets if breasts are large), then place one piece between parchment or plastic and gently pound with a meat mallet until about 1/4-inch thick.
Season both sides of each pounded chicken cutlet with coarse salt and freshly cracked pepper.
Set up three shallow bowls: one with the flour (seasoned with salt and pepper), one with the beaten eggs (seasoned with salt and pepper), and one with the bread crumbs (seasoned with salt and pepper).
Dredge each cutlet in the flour, shaking off excess, then dip it into the egg wash to coat fully, and finally press it into the bread crumbs until evenly coated; transfer to a parchment-lined sheet tray.
Pour oil into a large frying pan so it comes about halfway up the cutlets and heat over medium until it reaches about 375°F (or is hot and shimmering).
Fry the breaded cutlets in batches (2 at a time depending on pan size) for 2 to 3 minutes per side, or until golden brown and cooked through.
Transfer cooked cutlets to a rack set over a sheet tray and repeat with remaining cutlets, changing the oil if it becomes too dark from breadcrumbs.
Serve hot, optionally with parsley, grated Parmigiano, and lemon slices, or use in sandwiches.
