First, make the mushroom peperonata. Thinly slice the onion, garlic, pepper and mushrooms. Chop the tomatoes.
Heat the oil in a large frying pan and soften the onion, under a lid, for about 5 minutes.
Add the garlic, thyme leaves and pepper slices to the pan, and cook for 5 minutes over low heat. Add the tomatoes and mushrooms. Stir in the vinegar, add the chilli if used, then put the lid on.
Simmer over low heat, under cover, for 15-20minutes.
Meanwhile, wipe the steak with paper kitchen towel and rub with oil.
When the sauce has reduced, crumble the cheese on top and set aside (keep the lid on).
Heat a griddle pan, frying pan or BBQ. Season the steaks with salt and pepper.
When the pan is hot, seal the steak on both sides for 30 seconds, then cook for 3-5 minutes on each side, turning them every minute. The exact time depends on the thickness of the meat and your liking of beef steak.
Rest the steaks for 5 minutes, then serve them with the mushroom peperonata (reheat it first if needed), garnished with basil leaves.