In a small bowl, stir together the sour cream, applesauce, breadcrumbs, horseradish, 1/4 cup chives and a generous amount of pepper. Cover and refrigerate.
Meanwhile, in a large bowl, combine the beef, Worcestershire sauce and parsley; season with salt and pepper. Form into 4 patties and coat each with the EVOO. In a large skillet, cook the patties over high heat until the bottoms are well-browned, about 4 minutes; flip, lower the heat and cook for 4 more minutes for medium-rare.
Top each roll bottom with some of the watercress, a beef patty, lots of horseradish sauce, a sprinkling of chives and a roll top.