Place the potatoes in a sauce pot with water to cover by one inch. Bring to a boil over high heat, add some salt, and cook the potatoes until tender, 12 minutes. Turn off the heat. Drain the potatoes and return to the pot, and let them sit for a minute to dry out. Add the butter and a splash of milk or half-and-half. Add the egg and mash mixture with a potato masher or fork. Add the chives and salt and pepper and continue to mash until fairly smooth.
Preheat broiler or grill.
While the potatoes are cooking, start on the burgers. In a bowl, combine the sirloin, worcestershire, and parsley. Grate about 2-3 tablespoons of the onion directly into the bowl; wrap up the remainder of 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 into 4 equal portions. Form each portion into a patty about 1-inch thick and press a slight indentation into the center of each. This dent will promote more even cooking and prevent the burger from bulging.
Preheat a large skillet over medium-high heat with about 2 tablespoons of olive oil. 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-done, turning once. Use the same timing if cooking on a grill.
Split the kaiser buns and toast lightly under the broiler for just a minute, if desired.
To build the burger, slather the bun bottoms with mustard and top with watercress or lettuce. Place the burger next and the put a mound of the mashed potatoes on top of each. Put top of bun over potatoes and enjoy.