In a large, heavy skillet set over medium-high heat, cook the onion in a drizzle of oil for about 5 minutes, until soft and golden. Transfer to a bowl and set aside. Meanwhile, shape the beef into patties, sprinkle with salt, and cook in another drizzle of oil in the skillet for 4-5 minutes per side, or until done to your liking.
Spread slices of bread with soft butter or mayo, and make a sandwich with a burger patty, caramelized onions and cheese. Add a little more oil to the pan and cook the sandwiches until golden on both sides. If necessary, cover with a lid or piece of foil to help them heat through.