Put green beans, water and 1/4 teaspoon salt in a deep 12-inch skillet. Cover and steam over medium heat for 4 to 5 minutes until crisp-tender, then transfer to a covered bowl and wipe the skillet dry.
Shape beef gently into 4 oval patties about 3/4 inch thick. Season both sides with remaining salt and finely ground pepper.
Heat oil in the same skillet over medium heat. Cook patties for 5 to 6 minutes per side until browned and their centers reach 160°F, then transfer to a clean plate. Spoon off excess fat, leaving about 1 tablespoon in the skillet.
Cook shallot in the remaining drippings for 2 minutes. Stir in crushed peppercorns for 20 seconds, then add broth and scrape up the browned bits.
Stir in cream, Dijon and Worcestershire and simmer gently for 3 to 4 minutes until the sauce lightly coats a spoon. Return patties, plate juices and green beans and warm over low heat for 2 minutes.
Serve one hamburger steak with one-quarter of the beans and peppercorn cream per portion.