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Combine the beef, salt, bread and half of the onions in a large bowl until well mixed. Shape the meat mixture into four 4-by-3-inch oval patties. Place the patties on the broiler pan rack 6 inches from the heat. Broil until an instant-read thermometer inserted into the center registers 160 degrees F for medium, about 12 minutes, turning once halfway through cooking time.
Heat the oil in a medium nonstick skillet over medium-high heat. Add the mushrooms and the remaining chopped onions. Cook, stirring occasionally, until the mushrooms are tender, about 8 minutes. Stir in the flour and cook 1 minute, stirring constantly. Add the broth and Worcestershire and bring to a boil. Add the patties to the skillet and bring to a simmer and cook until the sauce bubbles and thickens and the patties are heated through, 2 to 3 minutes.