Fill a large container with 3 quarts of cool water. Whisk the salt and sugar into the water to dissolve it. Submerge the chicken in the brine and refrigerate for 30 minutes to 1 hour. Remove the chicken from the brine and pat dry with paper towels.

Heat oven to 325 F. Place the potatoes, shallots, 1 halved garlic head and 1 tablespoon of the olive oil in a large cast-iron skillet. Generously season with salt and black pepper and toss to combine. Set aside.

In a small bowl, combine the remaining 1 tablespoon of olive oil, lemon zest, 2 teaspoons of the rosemary, 2 teaspoons of the Dijon and ½ teaspoon of the black pepper.

Loosen the chicken skin from the meat and rub the zest mixture under the skin, coating as much of the meat as possible.

Place the remaining halved garlic head inside the chicken, then tie the legs together with butcher’s twine. Nestle the stuffed chicken between the vegetables.

In a small saucepan over medium heat, melt the butter with the remaining 1 teaspoon rosemary, 1 teaspoon Dijon and ½ teaspoon black pepper. Stir the lemon juice into the melted butter. Turn off the heat.

Brush the chicken and vegetables with some of the lemon butter. Transfer the skillet to the oven and bake. Continue to brush the chicken and vegetables with the remaining lemon butter every 30 minutes, until the chicken is golden brown and the internal temperature reaches 165 F on an instant-read thermometer, 2 to 2 ½ hours total.

Serve the chicken, vegetables and the pan sauce on a large serving platter. Garnish with chopped parsley and lemon wedges.