Whisk soy sauce, sugar, vinegar, ginger, pepper flakes, and cornstarch together in a bowl.
Pat steak dry with paper towels. Heat 1 tablespoon oil in 12-inch nonstick skillet over medium-high heat until just smoking. Cook steak, turning as needed, until well browned on both sides and meat registers 120-125 degrees (for medium-rare), 8-12 minutes. Transfer to cutting board and tent with aluminum foil.
Add remaining 1 tablespoon oil and scallions, mushrooms, and broccoli to skillet and cook over medium heat until lightly browned, 2-3 minutes; transfer to bowl.
Add soy sauce mixture and any accumulated meat juices to skillet and simmer until thickened, 2-3 minutes. Slice steak thinly on bias against grain. Transfer to platter and top with sauce, scallions, and sesame seeds. Serve.