Gently mix the beef, oregano, garlic and 1/2 teaspoon each salt and pepper in a medium bowl until just combined.
Shape the meat into 4 6-ounce patties, about 5-inches in diameter. Make a slight indentation in the center of each patty and lightly sprinkle the outside of the patties with salt and pepper. Lightly oil the grill grates. Grill to desired doneness, about 4 minutes per side for medium, adding a slice of cheese to each burger during the last minute of cooking.
Mix the mayonnaise and balsamic vinegar together in a small bowl. Quickly toast the buns on the grill for about 1 minute and then spread with some of the mayonnaise mixture. Top with a burger and then some fresh basil leaves. Serve with giardiniera on the side.
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