Preheat oven to 300 degrees.

Place potatoes in a large roasting pan or oven-safe skillet. Toss with 3 tablespoons olive oil and salt and pepper. (Season with an amount of salt and pepper to your preference. The measured salt and pepper amounts in the ingredient list are for the chicken.)

Remove giblets from chicken and set aside for another use or discard. Pat chicken dry with paper towels. Place chicken in pan with potatoes. Stuff the cavity of the chicken with lemon quarters and garlic cloves. Tie the legs together with cooking twine.

In a small bowl, mix together 3 tablespoons olive oil, 1 tablespoon kosher salt, brown sugar, 1 teaspoon pepper, and paprika. Rub mixture all over the outside of the chicken. (Lift up chicken to get the bottom of the thighs.)

Transfer to oven and cook chicken for about 3 hours total, until the skin is browned and the meat is very tender.

Meanwhile, toss carrots with 2 tablespoons olive oil and salt and pepper. Set aside in a smaller baking dish. Cut off the top 1/3 of the head of garlic and place garlic on a piece of aluminum foil. Drizzle the remaining 1 teaspoon olive oil over the cut top of garlic. Close foil over the top to cover.

After chicken has been roasting for 1 hour 45 minutes, spoon some of the pan juices over the chicken and turn over the potatoes. Add chicken back to oven and also put carrots into the oven. (The carrots need about 1 hour 15 minutes to bake.)

When chicken has 40 minutes left on timer, add garlic to the oven in the dish with the carrots. (The garlic needs about 40 minutes to bake.) Carrots and garlic are done when tender. The garlic cloves can be squeezed out of the peel once cool enough to handle.

Allow chicken to rest 10 minutes before carving. Serve chicken pieces with potatoes and carrots, and roasted garlic if you like.