In a small bowl, soak bread in broth. Tear with your hands until it resembles a soft paste.
In a large bowl, combine ground beef, soaked bread, onions, egg, salt, and pepper. Gently mix until combined.
Divide meat into portions and shape into patties. With your thumb, gently press in the center of each patty to make a slight indentation.
In a skillet over medium heat, heat oil. Add mushrooms and cook, stirring regularly, for about 1 to 2 minutes. With a slotted spoon, remove from pan and keep warm.
Add beef patties in a single layer and cook for about 1 to 2 minutes on each side or until lightly browned but not cooked through. Remove burgers from pan and drain on paper towels. Keep warm.
In a saucepan over medium heat, heat the butter until melted. Sprinkle in the flour and whisk until smooth and blended. Continue to cook, stirring regularly, for about 2 to 3 minutes or until roux turns lightly golden.
Slowly add beef stock, whisk until smooth, and continue to cook until sauce begins to thicken. Add salt and pepper for the gravy.
Add meat patties in a single layer in the pan. Lower heat, cover, and continue to cook for about 10 to 15 minutes or until meat is fully cooked.
Add mushrooms during the last 5 minutes and cook until heated through. Serve hot.