Heat broiler. Heat oil in large, heavy nonstick skillet over high heat. Add meat; saute 1.5 minutes per side. Remove from pan to cutting board.
Slice meat across grain at an angle. Transfer to plate; cover and keep warm. Add onion, salt and pepper to skillet. Reduce heat to medium; saute until onion is softened, about 10 minutes.
Place meat on bottom of buns, dividing equally. Divide onion over meat. Top with cheese. Place on baking sheet.
Broil sandwich bottoms until the chees melts, about 1 minute. (The sandwich tops, cut side up, may be toasted under the broiler at this time.) Place the tops on sandwiches. Serve immediately.