Warm a 12-inch skillet over medium-high heat. Season steaks with salt and pepper. Swirl 2 tsp. oil in skillet. Add steaks; cook until browned on one side, 3 to 4 minutes. Turn and cook, 3 to 4 minutes longer for medium-rare. Transfer to a platter and tent with foil to keep warm.
Add remaining 2 tsp. oil, white and light green scallion pieces and ginger to skillet; sauté for 1 minute. Add garlic; sauté for 30 seconds. Add sherry and broth; cook, stirring, until liquid has reduced by about half, 30 seconds to 1 minute. Stir in oyster sauce. Remove skillet from heat; add butter and swirl until sauce is creamy.
Slice steaks across grain; spoon sauce over. Top with scallion greens and sesame seeds and serve.