Pat steaks dry and cut 3 (1-inch-long) slits, 1-inch apart, across center cartilage (to keep meat from curling), then sprinkle steaks with salt and pepper.
Heat oil in a 12-inch heavy skillet over moderately high heat until hot but not smoking. Cook steaks, turning over once, until meat is just medium-rare, 4 to 6 minutes total. Transfer steaks to a platter and cover loosely with foil.
Add butter to skillet and saute mushrooms and shallot, stirring frequently, until mushrooms are browned and tender, about 4 minutes, then transfer to platter with steaks.
Add vinegar and soy sauce to skillet and simmer, stirring and scraping up any brown bits, 2 minutes. Add 1/2 cup broth and simmer 2 minutes.
While sauce simmers, stir cornstarch into remaining 2 teaspoons broth in a cup. Stir cornstarch mixture into sauce and simmer, stirring, 1 minute. Return steaks and mushrooms, along with any juices accumulated on platter, to skillet and simmer, turning steaks over in sauce, until just heated through, about 1 minute.
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