Place the chicken cutlets between two sheets of plastic wrap and gently pound to about 1/4 inch thick for even cooking.
Set up a breading station: flour seasoned with salt and pepper in one shallow dish, whisked eggs in a second, and a mixture of panko and grated parmesan in a third.
Dredge each cutlet first in the seasoned flour, tapping off any excess, then dip into the egg, and finally press into the parmesan-panko mixture until well coated.
Heat olive oil and butter in a large skillet over medium-high heat. When it shimmers, add the chicken and cook until golden and cooked through, about 2–3 minutes per side depending on thickness.
Transfer the cutlets to a plate and squeeze a little lemon juice over them while they rest briefly.
If using tomatoes, sauté them in the same pan for 1–2 minutes until slightly softened, then remove and set aside.
Assemble the arugula: toss arugula with olive oil, a pinch of salt, and a little lemon juice to taste. If you like oregano, sprinkle a pinch for aroma.
Plate the chicken over a bed of arugula. Scatter tomatoes if using, drizzle with the remaining lemon juice, and finish with extra parmesan or a light sprinkle of black pepper.