Trim off and discard any excess fat from steaks, then place the meat between two pieces of plastic wrap. With a heavy, flat-sided mallet or a clean hammer, firmly pound all over until the meat is 1/4 to 1/3 inch thick. (Don't pound hard enough to tear the meat.)
Melt butter and oil in a 10- to 12-inch nonstick frying pan over medium-high heat. When butter sizzles, add beef (if all the meat won't fit, cook in batches). Cook, turning once, until both sides are well browned and center is no longer pink (cut to test), 4 to 5 minutes total. Remove meat, transfer to a platter, and keep warm.
Smoothly blend broth, sherry, soy sauce, ginger, and cornstarch. Add mixture to pan and boil until slightly thickened; pour over meat. Drizzle sesame oil and sprinkle chopped green onion on top.