Trim visible fat from the sirloin and freeze it on a plate for 20 minutes, until firm but not frozen solid. While it chills, slice the bell peppers into 6 mm strips, the onion into half-moons and the mushrooms 5 mm thick, and mince the garlic.
Slice the firm beef against the grain as thin as you can, about 3 mm. Toss the slices with 1/2 tsp of the salt and the black pepper.
Heat 1 tbsp of the oil in a 30 cm / 12-inch cast-iron skillet over medium-high heat for 2 minutes, until it shimmers. Sear half the beef in a single layer for 1 to 2 minutes, stirring once, until browned with no pink left. Move it to a plate, then sear the second half the same way.
Add the remaining 1 tbsp oil to the skillet, then the onion, peppers and the remaining 1/4 tsp salt. Cook for 4 minutes, stirring every minute, then add the mushrooms and cook 5 minutes more. The pile will shrink by about half. Stop when the onions are soft and browned in spots and the mushroom liquid has cooked off, then stir in the garlic for 30 seconds.
Pour in the Worcestershire sauce and broth and scrape up the browned bits for 1 minute, until the liquid has reduced by half. Return the beef and its juices to the pan and stir for 1 minute over medium-low heat. Overlap the provolone slices across the top, cover with a lid or foil for 2 minutes until melted, and serve from the pan.