Partially debone the chicken halves, removing the wing, backbone and thighbone. Rinse and pat dry. Refrigerate, uncovered, at least 30 minutes to dry out the skin.
Season the chicken with salt and pepper on both sides. Heat a large nonstick skillet (preferably cast iron) over medium heat and add the oil and butter. Lay the chicken halves, skin-side down, in the skillet and scatter the thyme around them. Wrap a slightly smaller skillet with foil and place on top of the chicken. Set a large brick or 2 to 3 heavy cans in the skillet to weigh it down.
Cook the chicken 3 to 5 minutes, then remove the top skillet and check the skin; adjust the heat and rotate the pan as needed so the skin browns evenly. Replace the top skillet and continue to cook until the skin is golden and crisp and the chicken is cooked about three-quarters of the way through, 20 to 25 more minutes. Remove the top skillet and carefully flip the chicken. Cook, uncovered, until the chicken is cooked through, 8 to 10 more minutes. Transfer to a cutting board and let rest 5 minutes. Cut each piece in half to serve.