Place 1 tablespoon olive oil in a large resealable plastic bag and shake. Put 1 pork chop in the bag and seal; pound the meat with a mallet or heavy skillet, avoiding the bone, until it’s about 1/4 inch thick all over. Remove pork chop and repeat the process with the rest of the chops (no need to add more oil).
Combine arugula, cherry tomatoes and onion in a large bowl. Drizzle with vinegar and 4 tablespoons oil and toss well. Season with salt and pepper.
Oil grill grates and preheat grill to high. Season chops with salt and pepper, place on hot grill rack and cook 2 to 3 minutes on each side, until well browned.
Serve chops with salad piled on top. Use a vegetable peeler to shave thin slices of Parmesan over salad.