Pat the chicken completely dry and season generously on all sides with AP seasoning, Cajun seasoning, and black pepper. Place uncovered in the refrigerator for 12–24 hours to dry brine.

Remove the chicken from the refrigerator 30 minutes before cooking. Stuff the cavity with lemon halves, garlic, rosemary, and thyme, then rub the softened butter all over the skin.

Preheat the oven to 425°F. In a roasting pan, toss the potatoes, carrots, Brussels sprouts, and onion with olive oil, salt, pepper, and a light sprinkle of Cajun seasoning. Add garlic cloves, rosemary, and thyme, then pour the chicken stock into the bottom of the pan.

Place the chicken breast-side up on a rack over the vegetables. Roast at 425°F for 20 minutes, then reduce the oven temperature to 375°F and continue roasting for 45–60 minutes, until the internal temperature reaches 165°F in the thigh.

Remove the chicken from the oven and let rest for 15 minutes before carving. Spoon pan juices over the vegetables and serve.