Place the chicken between parchment or plastic wrap and pound until about ¼ inch thick. Season both sides generously with kosher salt and black pepper.
Set up three shallow bowls: flour in one, beaten eggs in another, and panko mixed with the grated Parmesan in the third.
Dredge each cutlet in flour, then egg, then coat thoroughly in the breadcrumb mixture, pressing lightly so the crumbs adhere.
Heat about ¼ inch of oil in a large skillet over medium-high heat. When the oil is hot and shimmering, cook the cutlets until deeply golden and crispy, about 2–3 minutes per side. Transfer to a rack or paper towel-lined plate and sprinkle lightly with salt.
In a large bowl, combine the snap peas, matchstick radishes, scallions, dill, mint, lemon zest, lemon juice, olive oil, and grated Parmesan. Season with flaky salt and black pepper and toss gently.
Place each crispy chicken cutlet on a plate and pile the herby snap pea crunch salad in the center, leaving the golden edges of the cutlet visible. Finish with a squeeze of lemon and serve immediately.