Make the masher: place the potatoes in a saucepan and cover by 1 inch with water.
Place over high heat and bring up to a bubble.
Add some salt and boil the potatoes until tender, about 12 minutes.
Turn off the heat.
Drain the potatoes and return to the pot; let them sit for a minute to dry out.
Add the cream, the butter, and egg.
Mash with a potato masher or fork.
Add the chives and season with salt and pepper; continue to mash to a fairly smooth consistency.
While the potatoes are cooking, start on the burgers—in a large bowl, combine the beef, Worcestershire sauce, parsley, and the grated onion (grate it right over the bowl so the juices fall into the meat); season with salt and pepper and mix thoroughly.
Score the mixture into 4 equal portions and form them into patties slightly thinner at the center that at the edges.
Heat the oil in a large skillet over med-high heat.
Cook the burgers, flipping once, 10 minutes for medium.
Slather the roll bottoms with mustard and top with the watercress, burgers, and a pile of mashers.
The mashers will be the glue to hold the roll tops in place.