Shape the ground beef into four doughnut-shaped ½-inch-thick patties (the hole in the center will allow the patties to cook evenly and will fill in as the patties shrink during cooking); sprinkle with salt and pepper, if desired.
Heat the oil in a large nonstick skillet over medium heat.
Dip each patty into the flour, making sure to coat both sides, then place the patties in the skillet.
Cook until browned, 3-5 minutes on each side.
Move the patties to one side of the skillet and add the onion on the other.
Cook the onions, stirring, until they are lightly browned, 2-3 minutes.
Spread the hamburger patties back out among the onions and pour the beef broth into the skillet.
Cover and continue cooking until the meat is no longer pink in the center, about 10 minutes; taste and adjust seasoning with salt and pepper; serve hot.