Chop the onion and add the onion to a small pot with 1 tbsp of butter per half an onion, 1 tbsp of water, and 1/4 tsp salt. Set on medium heat and cook, stirring for 2 minutes until the butter has melted. Cover the pot, turn the heat down to medium-low, and let it stew for about 10-12 minutes until soft and translucent.
Take 4 oz of ground beef and form a ball. Then, form the burger patty. You can do so by pressing with your hands. Or you can do so by pressing the ground beef ball between two pieces of non-stick paper/parchment paper using a pot or a pan or anything with a flat bottom.
Season the patty with salt and pepper (about 1/4 tsp salt and 1/4 tsp black pepper per patty).
To a cast iron pan over medium-high heat, add the oil, and once the oil is hot, add the burger patty and cook for about 3 minutes on each side.
Top with stewed onions and, optionally, cheese.
While the cheese is melting, toast the bun in the same pan as the burgers.
Add 2 tbsp of unsalted butter to the underside of the top bun, and add the buttered top bun to the burger.