Mix first 4 ingredients in large bowl until blended. Shape into 4 balls. Poke deep hole in each ball and fill each with 1 1/2 tablespoons of cheese. Mold meat around cheese to enclose. Flatten each filled burger to 3/4-inch-thick patty. (Can be made 8 hours ahead, covered and chilled.)
Grill burgers to desired doneness, about 5 minutes per side for medium.
Place burgers on bottom halves of buns. Top with lettuce, tomatoes, onions and upper halves of buns.
Peel onions, leaving root end intact. Slice into 3/4-inch thick rounds. Lay onion flat and insert a toothpick horizontally through at least 1/2 of the onion. This will prevent onion from falling apart on the grill. Brush with oil and season with salt. Place on a preheated grill at medium heat. Cook for 5 minutes on each side or until caramelized and softened. Remove toothpicks before serving.