Preheat oven to 375 F, with 1 rack in the center position and another 4 inches from broiler. Line a large baking sheet with foil.
Whisk together soy sauce, sesame oil, brown sugar, garlic, and ginger in a medium bowl. Measure out 1/4 cup of the marinade. Add steak to remainder, turning to coat.
Toss broccoli florets and mushrooms on prepared baking sheet with olive oil and reserved marinade. Spread vegetables out in an even layer, transfer to center rack in oven, and roast until just tender, about 10 minutes. Remove baking sheet from oven and set oven to broil.
Push broccoli and mushrooms to edges of baking sheet and place steak in the center, drizzling over any remaining marinade. Slide baking sheet under broiler and broil until steak begins to char on the outside and a thermometer inserted into thickest part registers 125 F for rare or 135 F for medium-rare, 3 to 5 minutes per side.
Remove baking sheet from oven and let steak rest, loosely covered with foil, for 10 minutes before thinly slicing against the grain. Serve warm, with vegetables alongside.
Adapted from One Pan & Done Copyright 2017 by Molly Gilbert. Published by Clarkson Potter, a division of Random House, LLC.
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