PREHEAT barbecue to medium-high. Heat a large non-stick frying pan over medium. Add 2 tsp oil, then onion. Cook, stirring, until softened, 4 to 5 min. Reduce heat to low, then add vinegar and sugar. Cook, stirring occasionally, until onion is dark golden and sticky, 18 to 20 min.
STIR beef, 2 tsp oregano, 1/2 tsp salt and half the garlic in a bowl. Season with pepper to taste. Shape into 4 patties, about 5 in. wide. Toss peas with remaining oil and salt, then place in a grill basket.
OIL grill. Grill burgers and peas, lid closed, until peas are charred and tender-crisp, 2 to 3 min. Remove peas to a bowl, then stir in remaining oregano. Flip burgers and continue grilling, lid closed, until cooked, 2 to 3 min.
STIR mayo and remaining garlic in a bowl. Spread over bottom buns. Top burgers with caramelized onions and serve with snap peas.