Spray a broiler pan rack with cooking spray. Preheat the broiler.
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.
Per serving: Calories 218; Total Fat 7.5 grams; Saturated Fat 2 grams; Protein 26 grams; Total Carbohydrate 13 grams; Sugar: 4 grams; Fiber 4 grams; Cholesterol 60 milligrams; Sodium 454 milligrams