Preheat oven to 450°. Place a large cast-iron or other heavy skillet over medium-high heat. Add mushrooms, onion and green pepper; cook and stir until tender, 8-10 minutes. Remove from skillet; cool slightly.
Cut steak into 4 pieces; pound with a meat mallet to 1/4-in. thickness. Spread with cheese. Sprinkle with pepper; top with mushroom mixture. Roll up from a short side; secure with toothpicks. Place back in skillet.
Bake until meat reaches desired doneness (for medium-rare, a thermometer should read 135°; medium, 140°, medium-well, 145°), 12-17 minutes. Let stand 5 minutes before serving. Serve with mashed potatoes.