Combine 1/4 cup minced chives, 2 teaspoons lemon rind, 3 tablespoons lemon juice, 1 teaspoon oil, salt, and 1/8 teaspoon hot sauce in a large zip-top plastic bag; add chicken to bag. Seal and marinate in refrigerator 2 hours. Remove chicken from bag, and pat dry; discard marinade.
Heat 2 teaspoons oil in a large skillet over medium heat. Add chicken, and cook for 3 minutes on each side. Reduce heat to medium-low. Add 2 teaspoons lemon rind, 3 tablespoons lemon juice, 1/8 teaspoon hot sauce, carrots, and broth, and simmer for 15 minutes or until carrots are tender. Spoon 1/3 cup sauce onto each of 4 plates, and arrange 1 chicken breast half and 8 carrots on top of sauce. Sprinkle each serving with 1 tablespoon minced chives. Garnish with additional fresh chives and lemon slices, if desired.