Bring water, sugar, and ½ cup salt to a boil in a 6- to 8-quart heavy pot, then reduce heat and simmer, uncovered, 15 minutes. Cool brine completely, then add chicken and soak, covered and chilled, 6 hours.

Remove chicken and pat dry.

Whisk together lime juice, fish sauce, garlic, mint, cilantro, red pepper flakes, and remaining teaspoon salt in a large bowl, then add oil in a stream, whisking until combined.

Open vents on bottom of grill and on lid. Light a large chimney starter of charcoal (80 to 100 briquettes). Leaving about one quarter of grill free of charcoal, bank lit charcoal across rest of grill so that coals are about three times higher on the opposite side.

When charcoal turns grayish white (after 15 to 20 minutes), check fire temperature and grill chicken parts according to instructions in accompanying article. Transfer chicken as cooked to bowl with vinaigrette and turn to coat, then transfer to a serving platter and keep warm, loosely covered with foil.

When chicken is almost done, grill lime halves, cut sides down, uncovered, over coals until grill marks appear, about 3 minutes, then transfer to platter with chicken. Serve limes with chicken and serve remaining vinaigrette on the side.

To make a Mediterranean vinaigrette for chicken , whisk together ¼ cup fresh lemon juice, 1 minced large garlic clove, 1 tablespoon chopped fresh rosemary, 1 teaspoon dried hot red pepper flakes, and 1 teaspoon kosher salt, then add ⅓ cup olive oil in a slow stream, whisking.