Preheat a griddle, large griddle pan or large cast-iron skillet over medium heat for at least 10 minutes.
Whisk together the barbecue sauce, mayonnaise, 2 tablespoons pepperoncini liquid, mustard, smoked paprika and vinegar in a small bowl. Season with salt and pepper.
Roughly chop the pepperoncini peppers and toss together with the coleslaw mix in a large bowl. Add three-quarters of the sauce and toss. Reserve the remaining sauce for the bun and burger.
Loosely form the ground beef into 4 balls. Season with salt and pepper on all sides.
Brush the griddle with canola oil and add 1 ball of beef griddle. Add 1/4 cup of the shaved onions on top of the ball and smash it down with a heavy metal spatula. You want to smash and spread the meatball, making sure those onions are pushed into the beef, and creating a patty that will be bigger than your bun. Cook until browned, about 2 minutes. Flip and cook until the onions are steamed and caramelized, about 2 minutes. For this burger, it’s OK to give it another press with your spatula—we want that Maillard reaction (see Cook's Note). Top the burger with 1 tablespoon of the remaining mayonnaise mixture, then 2 slices of cheese and cook until melted, 1 minute. Repeat with the remaining beef, cooking in 2 batches.
Meanwhile, spread the mayonnaise mixture on the bottom of the buns and cover with 4 dill pickle chips.
Once the cheese has melted, place each burger on a bottom bun and top with 1/4 cup of the slaw mixture. Smash each top bun on the slaw and enjoy! Your friends will thank you.
