Measure out 2 tablespoons of thyme leaves and set the remaining sprigs aside. In a bowl, combine the thyme leaves with the butter, grated clementine zest, and chopped garlic.

Gently slide your fingers under the turkey skin to separate the skin from the meat on the breast and legs. Rub three-quarters of the butter mixture under the skin. Spread the remaining butter mixture all over the outside of the skin and massage it in. Season the turkey with salt and pepper, set it in a roasting pan, and let it rest for at least 1 hour at room temperature.

Preheat the oven to 400°F (200°C/Gas Mark 6) and roast the turkey for 30 minutes, until lightly golden.

Carefully pour the broth into the roasting pan and roast for another 30 minutes.

Meanwhile, cut 5 of the garlic heads in half crosswise. When the 30 minutes are up, place the clementine halves, halved garlic heads, and the remaining thyme sprigs around the turkey in the pan. Roast for an additional 1 hour, basting every 20 minutes with the pan juices, until the skin is crisp and the turkey is cooked through (when you cut between the leg and thigh, the juices should run clear). If more time in the oven is needed, continue basting regularly.

Meanwhile, peel and crush the cloves of the remaining head of garlic. When the turkey is done, transfer the pan juices to a saucepan, add the garlic, and simmer for 5–7 minutes to reduce. Taste and adjust the seasoning if necessary.

Place the turkey on a serving dish and arrange the clementine halves and halved garlic heads around the meat. Serve with the sauce.