Mix flour, ¼ teaspoon salt and ¼ teaspoon pepper in zip lock bag, add ½ the chicken and close and shake to cover with flour mixture, remove and shake off excess flour back into the bag, add remaining chicken and repeat.
Heat oil in a large nonstick skillet over medium high heat, add chicken and cook 6 – 8 minutes turning once, until golden brown and juices run clear. Remove from pan and cut in half lengthwise, keep warm.
Place water, onion and garlic in same skillet, cook until onions are soft about three minutes, add tomatoes and remaining salt and pepper, cook until tomatoes are soft, stir in vinegar, then arugula, toss with tongs. Spoon over warm chicken and scatter cheese on top.