While stirring constantly, slowly pour in 2 cups beef broth and 1 teaspoon Worcestershire sauce. Bring to a simmer over medium heat, whisking often, 1 to 2 minutes. Continue to cook, whisking often, until thickened enough to coat the back of a spoon, 4 to 5 minutes. Return the hamburger patties to the skillet, nestle into the mushroom mixture, and cook until heated through, about 2 minutes.
