Preheat the broiler. Line a baking sheet with foil and evenly space the chicken cutlets in the center. Brush the chicken with 1 tablespoon olive oil and season with salt and pepper. Broil, rotating the baking sheet as needed so the chicken cooks evenly, until lightly browned, 2 to 3 minutes. Flip the chicken and continue to broil until cooked through, 2 to 3 more minutes.
Remove the baking sheet from the oven and top each piece of chicken with 1 slice ham and 2 slices mozzarella. Broil until the cheese melts, 1 to 2 minutes. Season with salt and drizzle with olive oil.
Meanwhile, cut the baguette crosswise into 4 equal pieces, then split open. Spread the bottom halves with butter and the top halves with mustard. Toss the watercress with the remaining 1/2 tablespoon olive oil and season with salt and pepper. Sandwich the chicken and watercress between the baguette halves.