If using fresh tomatoes, mark skins with an X and drop tomatoes into boiling water for 2-3 minutes. Cold shock them in ice water then peel and coarsely chop.
Heat grill or cast-iron skillet. Season the steaks with Kosher salt and black pepper.
Heat a deep skillet over medium-high heat with olive oil, 4 turns of the pan. Add mushrooms and brown them; add peppers, onions, garlic and oregano to pan, and season with salt and pepper. Partially cover pan and cook to tender, 7-10 minutes. Add sweet vermouth or wine to vegetables then add tomatoes. Cover and simmer to break down tomatoes, 15-20 minutes.
Grill or cook steaks on the grill or in the hot pan for about 8-10 minutes, turning once for medium-rare to medium doneness. Allow steaks to rest a few minutes then thinly slice the steaks against the grain and serve topped with lots of chunky mushroom, pepper and onion pizzaiola sauce.