Whisk the vinegar, sugar and 2 tablespoons vegetable oil in a large bowl. Add the coleslaw mix, giardiniera and scallions. Season with salt and pepper and toss.
Preheat a griddle or large skillet over medium-high heat. Add 2 tablespoons vegetable oil, the onion, bell pepper and a big pinch of salt. Cook, stirring occasionally, until the vegetables are browned and tender, about 7 minutes. Remove to a plate and wipe the skillet clean.
Form the beef into 4 small thick patties, about 2 inches wide and 3 inches tall. Add the remaining 2 tablespoons vegetable oil to the skillet. Add the patties and flatten using a metal spatula, pressing down hard until very thin. Season with salt and pepper and cook until browned and crisp around the edges, about 2 minutes. Flip, season with salt and pepper and top each with 2 slices cheese, then the onion and peppers. Continue cooking until the patties are just cooked through and the cheese is melted, about 1 more minute.
Transfer the patties to the buns. Serve with the slaw.