Peel onions and cut lengthwise into 1/2-inch-thick wedges. Place in a 12- by 17-inch roasting pan with 2-inch-tall sides. In a small bowl, mix oregano, 1 teaspoon salt, and pepper. Add half the oregano mixture and 1 tablespoon olive oil to onions; mix well. Set one or two wire racks (overlap, if necessary) over onions, extending beyond pan sides so racks are not resting directly on onions.
Remove giblets from chicken; discard or reserve for other uses. With poultry shears or sturdy scissors, cut along one side of the backbone of chicken. Pull bird open and lay, skin side up, on a flat surface; push down to crack bones slightly so bird lies reasonably flat. Pull off and discard lumps of fat. Rinse bird well on both sides; pat dry. Rub remaining 1 tablespoon oil and oregano mixture all over chicken. Set, skin side up, on racks over onions.
Roast in a 425° oven until a thermometer reaches 170° when inserted through thickest part of chicken breast to bone and 180° when inserted through thickest part of thigh at joint, 50 to 60 minutes. Set chicken on a board or rimmed platter and let rest 5 to 10 minutes. Remove racks from pan. If onions are not browned, stir and continue roasting until browned, 5 to 10 minutes longer. Scrape browned bits off pan bottom and stir into onions; spoon onions into a bowl.
Wrap tortillas in plastic wrap and heat in a microwave oven on full power (100%) just until hot and steamy, 45 seconds to 1 minute.
Serve chicken with onions, tortillas, and salsa. To eat, cut chicken off bone in chunks or slices and let guests wrap chicken in tortillas with onions, green salsa, and salt to taste.