Cook the onions while stirring occasionally for 2 minutes, then add a tablespoon of water and continue cooking and stirring while spreading the onions evenly on the pan. Keep adding water whenever the onions dry out and start sticking to the pan.
Season the beef with salt and pepper, and shape into balls and flatten them and season with salt and pepper again. Then in the same skillet melt 1 tablespoon of butter, and add the beef and flatten as you would do with a smashed burger.
Cook on one side for 2 minutes without turning to allow a brown crust to form, then turn and cook on the other side for 1-2 minutes. Remove onto a plate, and set aside. Wipe the skillet clean with paper towels.
Butter the other side, flip and cook for a minute then reduce the heat to medium-low and layer 2 slices of cheese followed by the beef patty, caramelized onions, 2 more slices of cheese, and cover with bread. Allow to cheese to melt. Remove onto a plate, slice in half and serve.