First, start the caramelized onions. Cut the White Onion (1) in half and then into quarters. Thinly slice into 1/4-inch (60mm) thick pieces.
In a medium saucepan over medium heat, melt the Butter (2 tablespoon). And add onions in gradually and cook for 2 minutes or until translucent, then season with Salt (1 pinch).
Continue cooking the onions on medium-low heat for 20 minutes, stirring occasionally and deglazing with a little Water (1 tablespoon) if the onions start to stick.
While the caramelized onions finish cooking, take the Ground Beef (1.5 pound) and lay on a large piece of plastic wrap, taking care to keep the grain of the ground beef intact on the plastic. Wrap up the meat and gently roll, forming a long sausage shape. Let firm up in the fridge for 10-20 minutes.
Once chilled, form your ground beef into 4.5oz/125g patties by slicing carefully with a serrated knife. Once the meat is cut, remove the plastic and use your finger to make an indent in the center of each patty (this slight depression in the beef will ensure even cooking).
Finish the caramelized onions and set aside.
Heat a cast iron skillet, a non-stick frying pan, or a griddle pan over high heat with Oil (2 tablespoon). Season the burgers with Salt (1 pinch) and then sear the burger patties for 3 minutes on each side. Look and feel for doneness.
Place a slice of Cheddar Cheese (6 slice) on top of each burger, then add a splash of water to the pan and cover with the lid. Set cooked patties aside to rest for around 2 minutes.
Add 1 tsp of Butter (2 tablespoon) to each of the Brioche Hamburger Bun (6) and toast lightly. To assemble the burgers: Spread toasted bun with mayonnaise, then a cooked burger patty with cheese. Top with caramelized onions, and then spread Ketchup (to taste) or Mayonnaise (to taste) on the top bun. Enjoy alone or with your favorite burger sides!