Place each chicken breast between two large pieces of plastic wrap and pound with a meat mallet or rolling pin until each is about 1/4 to 1/2 inch thick.
Season both sides of the chicken cutlets with coarse salt and black pepper.
Set up three shallow bowls: one with the flour mixed with a pinch of salt and pepper, one with the beaten eggs, and one with the breadcrumbs seasoned with salt and pepper.
Dredge each cutlet first in the flour, shaking off excess, then dip in the beaten eggs, and press into the breadcrumbs to fully coat; place breaded cutlets on a parchment-lined tray.
Heat about 1/2 cup oil in a large skillet over medium heat until it reaches 350°F or a breadcrumb sizzles when dropped in.
Fry the cutlets 1–2 at a time without crowding the pan for 3 to 4 minutes per side, or until golden brown and cooked through; transfer to paper towels to drain.
In a large bowl, toss together the arugula, optional microgreens, sliced cherry tomatoes, thinly sliced red onion, shaved Parmesan, and a pinch of salt and cracked pepper; keep chilled until serving.
Whisk the lemon juice and honey in a medium bowl, then slowly drizzle in the extra virgin olive oil while whisking until emulsified; season the dressing with salt and pepper to taste.
Serve each chicken cutlet topped with a generous portion of the arugula salad, garnish with chopped parsley if desired, and drizzle with the lemon vinaigrette.
