Combine the bread crumbs, cheese, nutmeg, parsley, and orange zest in a shallow bowl. Put the flour in another bowl and the eggs in a third bowl.
Cut the chicken breasts horizontally being sure not to cut all the way through to the other side so it will open like a book. Once the chicken is butterflied, lightly pound the chicken between waxed paper, parchment or plastic wrap into 4 large, very thin cutlets. Season the cutlets with salt and pepper and coat with the flour, then the egg and then the bread crumbs.
Heat a 1/2-inch of frying oil in very large skillet over medium to medium-high heat. When the oil is hot, add the cutlets 1 or 2 at a time, depending on the size of the pan. As cutlets are done, transfer to a baking rack to drain. Cutlets may be served room temperature but you want them to remain crisp.
While the cutlets cook, working next to the stove, peel and section the oranges and dice. Add them to a large bowl along with the tomatoes, oregano, if using, onion, extra-virgin olive oil and salt and pepper, to taste. Toss and when ready to serve fold in the arugula leaves.
Transfer the cutlets to serving plates and top with mounds of orange and arugula salad before serving.