1. Prepare chicken breasts by pounding each to an even thickness of about ½ inch, or slice into thin cutlets (~¼ inch) to ensure even cooking.
2. In a zip-lock bag or shallow dish, combine olive oil, garlic, herbs, salt, pepper, lemon zest or vinegar, and brown sugar if using. Mix well.
3. Add chicken to the marinade and massage to coat thoroughly. Seal and refrigerate for at least 4 hours or overnight for best flavor and tenderness. A minimum of 30 minutes to 2 hours can work if time is short.
4. Preheat grill to medium-high or high heat and oil the grates well to prevent sticking. If using a charcoal grill, wait until coals are covered with white ash.
5. Remove chicken from marinade, letting excess drip off (reserve some marinade for basting, if desired). Place chicken on the grill and cook covered for about 2 to 3 minutes per side for thin cutlets, or about 6 minutes per side for thicker breasts. Avoid overcooking—internal temperature should reach 165°F (74°C). Turn chicken carefully; if it sticks, wait 30 seconds before flipping. You may baste with reserved marinade during cooking.
6. After grilling, let chicken rest for 3 to 10 minutes to allow juices to redistribute and keep the meat moist.
7. Garnish with chopped fresh parsley before serving.