Melt 2 tablespoons butter in a large pot over medium heat. Add the onions, 2 tablespoons water and a big pinch of salt. Cook, stirring occasionally and adding 1 more tablespoon water if needed, until the onions are caramelized, 15 to 20 minutes. Stir in 2 more tablespoons water and 1 tablespoon Worcestershire sauce. Slowly add the cheese, stirring to melt. Cover and keep warm.
Meanwhile, combine the ketchup, mustard, steak sauce, mayonnaise, remaining 2 teaspoons Worcestershire sauce and a few grinds of pepper in a small bowl.
Heat a large cast-iron skillet over high heat. Season the steak generously with salt and pepper. Add to the skillet and cook until well browned, about 3 minutes per side for medium rare. Transfer to a cutting board and let rest 5 minutes.
Toast the bread and spread with the remaining 1 tablespoon butter; divide among plates. Thinly slice the steak and pile on top of the bread. Drizzle with all but 2 tablespoons of the ketchup mixture and scatter the onions on top.
Toss the reserved ketchup mixture with the watercress; season with salt and pepper. Serve with the sandwiches.