Shape the ground beef into 4 equal-sized patties, roughly 1/2 inch thick.

Score the top of each patty lightly with a knife in a cross-hatch pattern; this helps them keep their shape while frying.

Prepare a breading station: mix flour, salt, and pepper on one plate; beat the egg in a shallow bowl; and place breadcrumbs on a third plate.

Dredge each patty in the seasoned flour, dip into the beaten egg, and finally coat firmly with breadcrumbs on all sides.

In a large skillet, melt 1 tablespoon of butter over medium-low heat. Add the sliced onions and cook for 15-20 minutes, stirring often, until soft and golden brown. Remove onions from the pan and set aside.

Wipe out the skillet if necessary, then add the remaining 2 tablespoons of butter and the vegetable oil over medium heat.

Fry the breaded patties for about 4-5 minutes per side until the crust is deep golden brown and the meat is cooked through.

Add the cooked onions back into the pan around the patties and pour in the beef broth. Simmer for 1-2 minutes to deglaze the pan and warm the onions.

Serve the patties immediately, topped generously with the onions and pan juices.