Step 1Rinse pork but do not pat dry. Place 1 piece pork in a heavy duty resealable plastic bag and pound with rolling pin until about 1/4-inch thick. If using pork chop, be careful not to pound bone. Remove pork from bag and season with salt on both sides. Repeat with rest of pork.
Step 2Place flour in wide mound on dinner plate. Beat egg with water until smooth. Combine bread crumbs and cheese in large mixing bowl. Place arugula in salad bowl and dress with oil and lemon juice and season with salt.
Step 3Heat oil in large skillet over high heat. Roll 1 piece of pork with flour and shake off excess. Dip in egg bath and shake off excess. Place in bread crumbs and cover well. Press to firm coating, then remove and gently shake off excess. Repeat with remaining pork pieces.
Step 4Dip 1 corner of cutlet in oil. If oil is hot enough, pork will sizzle loudly. If this happens, add pork to pan and repeat with remaining pieces. If oil is too cool, wait until it is hot enough. Fry cutlets until well-browned on bottom, 2 to 3 minutes. Turn over, salt cooked side, and cook 1 to 2 minutes longer. Remove to paper towel to drain and salt second side. Repeat with remaining cutlets.
Step 5Place each cutlet on plate with best-browned side up, then place handful of dressed arugula on top. Serve immediately.