Heat a cast-iron skillet over high heat until very hot. Sprinkle both sides of the steaks with salt and pepper. Place the steaks on the hot skillet without overcrowding. Reduce the heat to medium and sear the steaks until well browned, about 4 minutes. Flip the steaks and cook until medium/medium-rare, about 6 more minutes. Set aside, loosely cover with foil and let rest before slicing.
Heat the oil over medium heat in a large nonstick skillet. Add the onions, season with salt and pepper and toss in the oil. Sweat the onions until soft and slightly golden in color, about 15 minutes. Set aside.
For the sandwich build: Thinly slice the steaks and trim off excess fat. Return the meat to the skillet over medium heat, in batches if necessary, creating a pile for each sandwich. Add the onions on top of the meat and then the shredded cheese on top of the onions. Add a roll on top of each pile to meld with the cheese and heat up the bun. Once the cheese is melted, using a spatula, remove the upside down sandwich and turn right side up. Add the giardiniera relish. DO NOT CUT and eat until gone.