Preheat your grill to medium-high. Combine the first 6 ingredients in a bowl. Combine the beef, egg, garlic, breadcrumbs, oregano, salt and pepper in a second bowl. Shape meat into four, 3/4" thick patties. Grill burgers for 4 minutes per side, or until entirely cooked through and centre registers 160 degrees F on an instant red meat thermometer.
While the burgers cook, grill the vegetables on the other side of the grill until just tender, about 2 minutes per side, and then set on a plate.
When cooked, top burgers with the cheese. Turn the heat to low, close the lid and cook 1 to 2 minutes more, or until cheese is melted.
Spread buns with the mayonnaise. Top the bottom buns with a lettuce leaf. Set on the burgers, grilled vegetables, top buns, and serve.