Break up beef in large bowl. Sprinkle feta, oregano, salt, and pepper over; toss. Divide into 8 portions; form each into 41/2-inch patty, pressing in center to form slight indentation. DO AHEAD Can be made 1 day ahead. Cover; chill. Puree garlic in processor. With machine running, gradually add 1/2 cup oil; process 30 seconds. DO AHEAD Can be made 1 day ahead. Cover; chill. Spray grill rack with nonstick spray. Prepare barbecue (medium-high heat). Brush onion with 1 tablespoon oil; sprinkle with salt and pepper. Place burgers and onions on grill. Cover; cook 5 minutes. Turn; cook until onions are charred in spots and burgers are medium-well, about 4 minutes longer.Transfer to plate. Brush cut sides of buns with garlic oil. Place buns, cut side down, on grill; toast about 1 minute. Turn; cook until heated, about 1 minute. Place burgers on bun bottoms. Top each with onion, tomato, and bun top.