Thinly slice the onions as close to paper-thin as possible; you should have about 3 cups.
Divide the ground beef into 4 loose 4-ounce balls; do not compact. Mix the salt and pepper in a small dish for even seasoning.
Heat a large cast-iron skillet or griddle over medium-high heat until very hot, 2–3 minutes. Spread the cut sides of the buns with the butter and toast on the griddle until golden, 1–2 minutes; set aside.
On the hot griddle, make 4 small mounds of sliced onions (about 3/4 cup each), spaced well apart.
Place a beef ball on each onion mound. Using a sturdy spatula or burger press, smash each ball firmly into a very thin patty about 1/4 inch thick, pressing the onions into the beef. Sprinkle each patty evenly with the salt-pepper mixture.
Cook undisturbed until the edges are deeply browned and the onions at the edges begin to blister, 2–3 minutes.
Scrape under each patty and flip so the onions are now against the griddle. Add the water to the pan and immediately cover with a lid or inverted sheet pan to trap steam. Cook until the onions are tender and the beef is cooked through, 1–2 minutes.
Uncover and top each patty with a slice of American cheese. Cover again briefly just until the cheese melts, 20–30 seconds.
Spread yellow mustard on the toasted buns and add dill pickle chips to taste. Place each patty on a bun and serve hot.