Start by seasoning the chicken cutlets generously with salt and pepper on both sides. Set up a breading station with three shallow large bowls or plates: one with flour, one with beaten eggs, and one with panko breadcrumbs mixed with 2 generous pinches of salt.
Dredge each chicken cutlet in flour (if using), dip it into the eggs, and then coat thoroughly in the panko mixture, pressing gently to adhere. Place onto a sheet pan or plate and set aside.
In a large bowl, combine the arugula, sliced fennel, orange segments, Castelvetrano olives, and pistachios.
In a small bowl or jar, whisk together the lime juice, orange juice, minced shallots, olive oil, and red wine vinegar or apple cider vinegar. Season with salt and pepper to taste.
Heat the oil in a large skillet over medium heat until shimmering. Fry the breaded chicken cutlets for 3-5 minutes on each side, or until golden brown and cooked through (internal temperature of 165°F). Transfer the cooked cutlets to a paper towel-lined plate to drain any excess oil.
Arrange the crispy chicken cutlets on plates and top each with a generous portion of the citrus salad. Drizzle the salad with the vinaigrette and garnish with fennel fronds for a touch of elegance. Serve immediately.