Place the potatoes in a sauce pot and cover by one inch with water. Place it over high heat and bring up to a bubble, add some salt and boil the potatoes until tender, 12 minutes. Turn off heat. Drain and return the potatoes to the pot and let them sit for a minute to dry out. Add the butter and milk or half and half just a splash. Beat egg and add and mash with potato masher or fork. Add in chives and salt and pepper and continue to mash to fairly smooth consistency. Preheat broiler.
While potatoes are cooking, start on the burgers. In a bowl, combine the sirloin, Worcestershire, parsley and about 2-3 tablespoons of grated onion. Wrap up remainder of the onion and save for another use. Season the meat with salt and pepper and mix to combine. Score the meat with the side of your hand and score 4 equal portions. Form the patties making them about 1 inch thick and press a light indentation in the center of each burger. This dent will prevent the burger from bulging and will promote more even cooking.
Preheat a large skillet over medium high heat with 2 turns of the pan of EVOO, about 2 tablespoons. Once the oil ripples, add the patties and cook on each side for 7-8 minutes for rare to medium rare, 11-12 minutes for medium to medium well doneness, turning once.
While the burgers are working, split the buns and lightly toast under the broiler.
To build your burger, slather the bun bottoms with Dijon mustard, top with watercress, burgers and a pile of mashers on top of burgers. The mashers will be the glue to hold bun tops in place.