Season steaks with salt and pepper; let stand 10 minutes. Heat a large skillet over medium-high; swirl in 2 teaspoons oil. Cook steaks, turning a few times, until browned all over, 8 to 10 minutes for medium-rare. Transfer to a plate; top each with a cheese slice.
Lower heat to medium. Add 2 tablespoons oil, chopped onion, and garlic to skillet; cook 1 minute. Add vinegar; bring to a simmer, scraping up browned bits. Stir in chopped tomatoes; simmer until they release their juices, about 5 minutes. Stir in remaining 1/4 cup oil and juices from steak plate; strain through a sieve. Whisk in mustard and oregano; season with salt and pepper. Season lettuce and tomato and onion slices with salt and pepper, drizzle with vinaigrette, and top with oregano leaves; serve, with steaks.