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1. 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.
2. 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.
3. 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.
4. Assemble burgers, topping them with onion,
arugula, and a hefty dollop of mayonnaise mixture.
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