The night before you plan to roast the chicken, place the chicken on a clean cutting board and flip it so the backbone is facing up. Using a sharp chef’s knife or kitchen shears, remove the backbone by making a cut along either side of the spine. Discard the backbone or save it in the freezer to use for making chicken stock. Flip the chicken over so the breast is facing up. Place your hands in the center of the breast and press until you hear a gentle pop. Pat the spatchcocked chicken dry with paper towels, then sprinkle salt all over the bird (see notes for salting guidelines). Use your hands to rub the salt into and under the skin. Place the salted bird on a baking sheet and transfer it to the fridge overnight to dry brine in the salt.
The next day, preheat the oven to 425°F. Peel the carrots and the shallots. Slice the peeled shallots in half lengthwise and set aside. Chop the peeled carrots on the diagonal into large pieces. Lastly, scrub the potato, and then, leaving the skin on, chop it into large cubes.
Remove the dry-brined chicken from the fridge and pat it all over with clean paper towels to remove any moisture. Season the front and back of the bird with some salt and fresh black pepper.
Heat a large cast iron pan over medium heat. Once it is hot, add a tablespoon of olive oil and swirl the pan around to coat. Place the chicken in the hot pan breast-side-down. Sear for 5 minutes. Flip the chicken and sear for a further 4 to 5 minutes. Transfer the chicken to a plate and set aside.
Add the halved shallots to the pan cut-side-down. Sear for 4 to 5 minutes, or until the cut sides of the shallots are golden brown.
Add the carrots, potatoes, fresh thyme, and fresh rosemary to the pan with a drizzle of olive oil. Season with salt and fresh black pepper, then toss the vegetables well to coat. Place the seared chicken back into the pan on top of the vegetables.
Roast the chicken and vegetables in the oven for about 40 to 45 minutes, or until the internal temperature reads 155°F. Transfer the chicken to a cutting board and the roasted vegetables to a serving dish. Allow the chicken to rest at room temperature for 10 to 15 minutes before carving and serving.
While the chicken is resting, make the pan sauce. Add 50 grams of butter to the chicken's roasting pan. Heat over medium-low heat until melted. Add 1 cup of chicken broth to the pan and stir. Bring to a boil, then reduce the temperature to a simmer. Allow the sauce to reduce for about 10 minutes.
In a small bowl, whisk together 1 teaspoon of cornstarch with 1 tablespoon of water to create a slurry to help thicken the sauce. Pour the slurry into the sauce, stirring continuously. Cook for a further minute or two until the sauce has thickened. Taste and adjust the seasoning to your liking.
Serve the chicken and roasted root vegetables warm with the buttery pan sauce.