Preheat the griddle or cast-iron skillet over medium-high heat until the cooking surface reaches approximately 425°F.
Divide the ground chuck into 8 loose, equal 4-ounce balls. Do not tightly pack or flatten the beef.
Slice the yellow onions as thinly as possible using a mandoline slicer or sharp knife. Separate the slices into loose piles.
Lightly butter the cut sides of the brioche buns. Toast them cut-side down on the griddle until golden brown, about 1 to 2 minutes. Remove and set aside.
Place 8 equal piles of sliced onions on the hot griddle, leaving enough space between each pile to smash the burgers.
Place one beef ball directly on top of each onion pile.
Cover each beef ball with a square of parchment paper. Press down firmly with a burger press or sturdy metal spatula until the patty is very thin and the onions are pressed into the beef.
Remove the parchment paper. Season the exposed side of each patty with kosher salt and black pepper.
Cook without moving the patties for 2 to 3 minutes, or until the edges are deeply browned and crispy.
Use a sharp metal spatula or griddle scraper to firmly scrape underneath each patty, keeping the browned crust and onions attached. Flip each patty once.
Immediately place one slice of American cheese on each patty. Cook for another 1 to 2 minutes, until the onions are tender, the cheese is melted, and the beef reaches an internal temperature of 160°F.
Stack two patties together for each burger.
Place the stacked patties on the toasted buns. Add dill pickles, mustard, mayonnaise, or burger sauce as desired. Serve immediately.