Slice each chicken breast in half widthwise to create two thinner cutlets.
Place one cutlet between two pieces of parchment paper or plastic wrap and gently pound with a meat mallet until about 1/4 inch thick; repeat with remaining cutlets.
Season both sides of each cutlet with coarse salt and freshly cracked black pepper.
Set up three shallow bowls: one with the all-purpose flour (seasoned with salt and pepper), one with the whisked eggs (seasoned), and one with the bread crumbs (seasoned with salt and pepper).
Dredge a cutlet in flour, shaking off excess, then dip into the egg wash to coat completely, and finally press into the bread crumbs so they adhere; place on a parchment-lined sheet tray. Repeat for all cutlets.
Heat oil in a large frying pan over medium heat until it reaches about 375°F (or hot enough that a breadcrumb sizzles on contact).
Fry cutlets two at a time (not overcrowding) for 2 to 3 minutes per side, until golden brown and cooked through.
Transfer cooked cutlets to a wire rack set over a sheet tray to drain and keep warm while finishing the rest.
Serve warm, optionally with lemon slices, grated Parmigiano, or parsley, or use in sandwiches as desired.