Heat grill to medium (350° to 450°). Put onion in a small bowl and toss with vinegar and 1 tbsp. oil; set aside. Mix giardiniera with mayonnaise and oregano; set aside.
Season beef with salt and pepper, mixing with your hands until just combined. Divide into 4 portions and form into patties. Using your thumb, press a dent into the center of each patty (it will even out as the burger cooks). Brush patties lightly on both sides with 1 tbsp. oil. Brush cut sides of buns with remaining 2 tbsp. oil.
Drain onion and discard marinade. Grill onion and burgers until onion is softened and charred slightly, about 5 minutes, and until burgers are done the way you like, about 8 minutes for medium-rare. In the last 2 to 3 minutes of cooking, top burgers with cheese and lightly toast buns.
Assemble burgers, topping them with onion, arugula, and a hefty dollop of mayonnaise mixture.
*Find giardiniera in the pickle aisle of well-stocked grocery stores.