Melt 1 tablespoon of butter in a large skillet over medium heat. Add the sliced yellow onions and brown sugar, stirring to coat the onions evenly.

Cook the onions, stirring frequently, for about 15 to 20 minutes until they are deeply golden brown, soft, and sweet. Transfer the caramelized onions to a plate and cover with foil to keep them warm.

While the onions cook, combine the lean ground beef, egg, breadcrumbs, salt, and black pepper in a medium bowl. Mix gently with your hands just until combined to keep the patties tender.

Divide the meat mixture into 4 equal portions and shape them into oval or round patties, about 3/4-inch thick. Press a slight indent in the center of each patty to prevent them from bulging while cooking.

Wipe the skillet clean, then melt the remaining 1 tablespoon of butter over medium-high heat. Add the beef patties to the pan and cook for 4 to 5 minutes on the first side until a rich, brown crust forms.

Flip the patties and cook for another 4 minutes, or until the internal temperature reaches 160 degrees Fahrenheit (71 degrees Celsius) on a meat thermometer. Transfer the cooked patties to a warm serving platter.

Pour the beef broth into the hot skillet, scraping up any flavorful browned bits from the bottom of the pan with a wooden spoon. Simmer for 2 minutes until the liquid reduces slightly into a light pan juice.

Pour the pan juices over the beef patties, top each patty generously with the warm caramelized onions, and serve immediately alongside mashed potatoes and steamed green peas.