Two to three hours before cooking, season the chicken with salt and refrigerate. Bring to room temperature before preparing.
Heat 1 tablespoon of olive oil and garlic in a sauté pan over medium-low heat for approximately 15 to 20 minutes. Stir the garlic to keep it from burning.
When the garlic is soft, heat the rest of the olive oil and 2 tablespoons of butter in a Dutch oven. Put the chicken in the Dutch oven and cook, skin side down, until the skin is crisp and golden. Turn them over and decrease the heat to medium-low.
After you have flipped the chicken, put the olive oil sauce and garlic in the chicken pan. Add half of the lemon juice, the white wine, the chicken stock, the thyme, and half of the lemon zest. Bring to a simmer and continue to cook, covered, for about 5 to 7 minutes, or until the chicken is tender and cooked thoroughly. Stir as needed and add water, if needed, to keep half an inch of liquid in the pan.
Remove the chicken, increase the heat, and bring the liquid to a boil. Remove the heat and add the rest of the butter. Season with salt, pepper, and the remaining lemon zest.
Pour the sauce over the chicken and decorate with the rest of the lemon zest. Be sure to discard the skin before eating.