Using a mandolin, slice onion very thin, so it's almost translucent. Divide meat into 4 even balls.
Heat a large frying pan over medium high heat for a few minutes. When it's hot, add the oil. When it's hot, place a meatball down. Top with a generous heaping of sliced onions (roughly ½ - ⅔ cup).
Use a press or pot lid and smush down. Hold for 20 seconds. Release and season with salt and pepper. Cook until edges get crispy, about 2 minutes.
Carefully flip the burger. Some onions may be stuck, and that's ok. Scrape carefully to get all the bits. Once flipped, press down on the patty to sink the onions into the meat. Season with salt and pepper and add a slice of cheese.
Place the top bun on top of the cheese, and the bottom bun on top of that. (This will allow the buns to steam as the burger finishes cooking). Cook for 2 more minutes.
Carefully scrape the burger up off the skillet. Grab the bottom bun, place under, and slide the burger and top bun onto it. Repeat 3 more times with remaining patties.