Preheat the oven to 425°F.
Blot the brined chicken dry with a paper towel.
With a sharp pair of kitchen shears, cut out the back bone and trim away excess fat and skin.
Then, spatchcock the bird: Flip the chicken over and open it up like a book. Using a kitchen knife, cut a ½-inch through the cartilage on the breast bone and firmly press down with your hands to flatten it.
Cut the leek in half lengthwise. Give the leek a quarter-turn, and then slice it lengthwise again (at a right angle from the initial cut). Keep the root end intact!
Fan the leaves under running water to release the sand and mud.
Once the leeks are cleaned, flick off the excess water and slice ’em crosswise.
Chop the carrots in uniform medium chunks and toss them on a tray with the leeks.
Add the quartered mushrooms and toss on some salt and pepper.
Dot the vegetables with half of the softened butter or ghee (2 tablespoons).
Lay the chicken on top of the seasoned vegetables, breast-side up.
Smush on the remaining two tablespoons of butter or ghee all over the chicken skin.
Roast the chicken in the oven for 45 minutes or until the breast registers 150° F on a meat thermometer and the thigh meat hits 170° F.
Transfer the chicken to a cutting board and let it rest for 15 minutes before carving it. In the meantime, toss the vegetables in the pan with a splash of balsamic vinegar.