Onion slicing: using a Japanese or other type of mandoline slicer, slice 1/4 of an onion into very thin slices. You want the slices to be almost paper-thin.
Cheeseburger patty: weigh your ground beef into a 4-ounce portion and then shape it into a ball. If you do not own a scale, you should be able to get 4 four-ounce-ish balls out of one pound of ground beef.
Preheat a griddle or pan to at least 400 F (205 C) for around 5 to 10 minutes.
Add the beef ball to the hot surface, top with all of the paper-thin slices of onion, and smash everything down as flat as you can on the griddle or pan. You're going to want to use a lot of pressure and smash hard. The goal here is to get as much surface area of the meat to touch the hot surface as possible. If you have a flat top or a griddle, you might want to use some sort of very heavy spatula or even a clean finishing trowel from your local hardware store.
As soon as you've finished smashing, season the top of the onion with a pinch of salt and black pepper. You will season again when you flip, but this is when you should do most of the seasoning.
Cook for 2 minutes on this first side or until the very edges of the onions that are touching the pan and the patty start to turn brown.
Use a sharp spatula and scrape under the patty to flip. The goal here is to make sure you don't lose any of the crispy brown bits you worked hard to build up with the smashing process.
Once flipped the patty will be resting on top of a bed of onions, season the top of the beef with a small pinch of salt and pepper and add one slice of American cheese on top of the patty. Place the top bun on top of the cheese so that it will get some steam while the beef is cooking and the cheese is melting.
Cook the patty on top of the onions for 3 to 4 minutes or until the onions under the patty are browning well. At this point, you can add condiments and/or pickle slices to the bottom part of the bun if desired.
Burger construction: using a spatula, scrape up and lift the bun and cheese-topped patty off of the griddle or pan, attempting to get as many onions off of the hot surface as possible.
Place the burger on top of the bottom bun to complete the fried-onion burger.