Use a mandoline or a very sharp knife to slice the onions as thin as possible. They should be translucent. This is the secret to getting them to caramelize and crisp up in the short time it takes to cook the burger.
Do not form patties. Instead, divide the beef into 4 loose, 4-ounce balls. This prevents overworking the meat and ensures a tender interior.
Heat a heavy cast iron skillet or griddle over high heat until it just begins to smoke. Place the beef balls on the griddle and immediately top each with a massive handful of the sliced onions—more than you think you need.
Using a heavy metal spatula (and a second tool to press down if needed), smash the beef and onions together with significant force. The beef should be flattened thin, and the onions should be pressed deep into the meat. Season the top with salt, pepper, and smoked paprika.
Squirt a small amount of yellow mustard over the raw side of the beef. Once the bottom is deeply browned and crispy (about 2 minutes), flip the burger so the onions are now directly on the hot surface.
Place a slice of cheese on the beef patty, then place the top bun directly on the cheese and the bottom bun (face down) on top of that. This allows the steam from the frying onions to soften the buns and infuse them with flavor.
Once the onions are charred and soft (about 2 more minutes), use the spatula to lift the entire stack onto a plate. Add pickles and serve immediately.