Position the top oven rack in the middle of the oven and preheat the oven to 425F (220C).
Remove any gizzards and the neck from the cavity of the chicken. Pat it dry with a paper towel and remove any excess fat or skin from the chicken -this last step is optional and it depends on the chicken, but do not trim too much to expose meat from underneath.
Season the chicken:
Generously season the outside skin and the inside of the chicken with salt and pepper. Drizzle 1 to 2 tablespoons of olive oil (or melted butter) over the top of the chicken and use a basting brush or your hands to distribute it all around the skin.
Stuff the chicken with the rosemary sprigs, the lemon halves, and the garlic cloves. Place the chicken inside a 9x13 baking dish (with sides).
Add the vegetables:
Chop the carrots and slice the Brussels sprouts in half. Add them into a large bowl, drizzle with about a tablespoon olive oil, season with salt and pepper, and toss to coat all the vegetables with the seasoning. Transfer the vegetables to the baking dish and distribute them around the chicken.
Roast:
Roast the chicken in the preheated oven for a total of 1 hour and 15 minutes. After 50 minutes, if the chicken is getting too browned, cover with foil for the remainder of the cooking time. Check for doneness around the breast at 165F (75C) and 175F (80C) around the thigh after the 1-hour mark.
Remove it from the oven and allow it to rest for 5 minutes. Transfer the chicken onto a large cutting board to slice it and serve it with the roasted vegetables.