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In a medium bowl, gently mix together beef with vinegar, making sure to mix only as long as it takes to just combine them (overmixing the meat will cause it to form a tighter, more meatloaf-like texture, so make sure not to do it any more than necessary). Form 2 patties slightly wider than the width of the buns, press a dimple in the center of each with your thumb, and season all over with salt. Refrigerate for at least 30 minutes and up to 1 hour.
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Meanwhile, in a small bowl, stir together olives and mayonnaise. Refrigerate until ready to use.
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Light one chimney full of charcoal. When all the charcoal is lit and covered with gray ash, pour out and spread the coals evenly over half of coal grate. Alternatively, set half the burners of a gas grill to high heat. Set cooking grate in place, cover grill and allow to preheat for 5 minutes.
Clean and
oil the grilling grate.
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Place patties directly over hot coals and cook, turning occasionally, until well charred and center of burgers register 110°F (43°C) on an instant-read thermometer, about 7 minutes. Place 2 slices of cheese on top of each patty and continue to cook until cheese is melted and burgers register 125°F (51°C) for medium rare or 135°F (57°C) for medium, 1 to 2 minutes longer. Transfer burgers to a large plate and let rest 5 minutes. (Move burgers to cooler side of grill at any time if they threaten to burn but haven’t reached their final internal temperature yet.)
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Toast buns over coals. Spread olive mayonnaise on top and bottom halves of each bun. Set patties on bottom buns, close burgers, and serve right away.
