Pat the chicken dry with a paper towel and let it sit at room temperature for 20 minutes while the oven preheats.
Combine 2 tablespoons of the olive oil, butter, garlic, 1 1½ teaspoons salt, chopped thyme and chili in a small bowl to make a creamy paste. Using your fingers, carefully loosen the skin on the breasts and rub some of the mixture in between the flesh and the skin. Rub the remaining paste all over the outside of the chicken, including a bit in the cavity. Season the skin with the remaining 1 ½ teaspoons salt.
Heat a 10-12-inch cast iron skillet or heavy-duty ovenproof pan over medium heat until a drop of water sizzles and instantly evaporates. Add the remaining tablespoon olive oil, then lay the chicken in the pan, skin side down.
Cook 10 minutes without moving the chicken. The skin should start to turn golden brown - adjust the heat so the pan doesn't become too hot and burn the skin.
Transfer chicken to the oven and roast 30 minutes. Carefully turn the chicken over and roast an additional 15-20 minutes, until the legs wiggle easily, the juices run clear and the internal temperature reads 160 degrees. Let chicken rest at least 10 minutes before carving — remember that it will continue to cook as it sits. Carve into pieces and serve on a platter on a bed of the parsley.
To remove the backbone from a chicken:
Put the chicken on a cutting board, breast side down.
Cut along the backbone on either side with a pair of scissors or a sharp knife.
Remove the bone and discard, or save it to add to homemade chicken stock