In a large pot, stir the salt, sugar and 4 cups water until dissolved. Add 4 cups cold water and the chicken. Cover and refrigerate 4 to 12 hours. Drain.
Position a rack in the bottom third of the oven; preheat to 325 degrees . In a large cast-iron skillet, heat the vegetable oil over medium. Pat the chicken dry; season lightly with salt. Add chicken, skin side down, to the skillet; cook 1 minute. Add the garlic, cut side down, and the thyme and rosemary. Cook until aromatic, 4 to 5 minutes. Place the herbs and garlic on top of the chicken to prevent them from burning and continue cooking until the skin is very crispy, about 3 minutes. Place 1 butter cube on each piece of chicken.
Transfer the chicken to the oven and roast 30 minutes. Increase the heat to 475 degrees . Turn the chicken; roast, watching closely to make sure it doesn't burn, until the skin is deeply browned and the chicken is cooked through, about 10 minutes more. Transfer the chicken, skin side up, to a wire rack set in a rimmed sheet pan and let rest 10 minutes.
Squeeze the cloves from the head of garlic onto the chicken. Strain the juices from the skillet into a small saucepan. Bring to a simmer over medium. Remove from the heat, whisk in the lemon juice and olive oil; serve with the chicken.