Preheat broiler.
Toast the bread and spread one side of each with mustard. Top the mustard side with 1 slice of ham and 1 slice of cheese. Place all on a baking sheet.
In a large pan filed halfway with water bring to a simmer over medium heat. Add the vinegar. Gently add each egg to the water, see prep tip. Cook for 3-5 minutes, until desired consistency. If you like your egg runnier cook for the shorter time.
In the meantime, place the baking pant of open-faced sandwiches in the oven and broil until cheese is melted. Keep an eye on it so it doesn’t burn. Remove from oven, plate each.
In a bowl and the greens and tomatoes. Toss with 2 tablespoons dressing. Divide the salad and top each open-faced sandwich with the salad.
Top each salad with 1 poached egg.
Serve at once with a fork and knife.