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Whisk white wine, oil, oregano, pesto, lemon zest and juice, garlic, salt, and pepper together for marinade in a small bowl.
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Put the chicken leg quarters in a resealable plastic bag and pour marinade over chicken. Seal the bag and shake to completely cover. Place in the refrigerator to marinate for at least 2 hours, turning over after 1 hour to ensure chicken quarters marinate evenly.
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Preheat the oven to 425 degrees F (220 degrees C). Bring chicken to room temperature while the oven heats.
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Drain marinade from chicken and pat dry; place in a baking dish.
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Roast in the preheated oven for 20 minutes. Turn chicken and cook until no longer pink at the bone and the juices run clear, about 20 minutes more. An instant-read thermometer inserted near the bone of a quarter should read 165 degrees F (74 degrees C).
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Meanwhile, during the last 20 minutes of baking the chicken, bring goat cheese to room temperature and make sauce.
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Heat butter in a skillet over medium heat. Saute onion and garlic in the hot pan until soft and translucent, about 5 minutes. Pour in white wine and lemon juice. Add sun-dried tomatoes and pesto and season with salt and pepper. Reduce heat and let simmer until slightly thickened, about 15 minutes.
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Remove chicken from the oven and top each chicken quarter with 2 slices of goat cheese, 1 on each thigh and 1 on each leg. Cover with equal amounts of sauce and serve immediately.