Plan ahead: The chicken needs to marinate for at least an hour. Do not marinate it longer than 12 hours or it will be too salty. Use a rasp-style grater to zest the citrus. The chicken takes longer to cook on a gas grill, so begin checking it at the end of the range in step 3.

Grate 1 teaspoon zest from orange, 2 teaspoons zest from lemon, and 2 teaspoons zest from lime. Place all citrus zest in bowl. Halve orange, lemon, and lime and set aside. Add 3 tablespoons oil, water, two-thirds of garlic, 2 teaspoons sugar, 1 ½ teaspoons salt, and ½ teaspoon pepper to citrus zest and whisk to combine. Transfer brinerade to 1-gallon zipper-lock bag. Add chicken, press out air, seal bag, and turn to distribute brinerade. Refrigerate for at least 1 hour or up to 12 hours.

FOR A CHARCOAL GRILL: Open bottom vent completely. Light large chimney starter filled with charcoal briquettes (6 quarts). When top coals are partially covered with ash, pour evenly over half of grill. Set cooking grate in place, cover, and open lid vent completely. Heat grill until hot, about 5 minutes.

FOR A GAS GRILL: Turn all burners to high, cover, and heat grill until hot, about 15 minutes. Leave primary burner on high and turn off other burner(s).

Clean and oil cooking grate. Place chicken skinned side down on cooler side of grill, with thicker ends facing coals. (Edges of chicken should be no more than 4 inches from center of primary burner if using gas.) Cover with disposable pan (if using gas, close lid) and cook until bottom of chicken just begins to develop light grill marks and is no longer pink, 6 to 9 minutes.

Flip chicken and rotate so that thinner ends face coals. Cover as before and continue to cook until chicken registers 140 degrees, 6 to 9 minutes longer.

Remove disposable pan and transfer chicken and citrus halves, cut side down, to hotter side of grill. Cook chicken, uncovered (covered if using gas), until dark grill marks appear, 2 to 4 minutes. Flip chicken and cook, uncovered (covered if using gas), until marked on second side and chicken registers 160 degrees, 2 to 4 minutes longer. Cook citrus halves until lightly charred, about 4 minutes.

Transfer chicken to cutting board, tent with aluminum foil, and let rest for 5 minutes. Squeeze 3 tablespoons orange juice, 1 tablespoon lemon juice, and 1 tablespoon lime juice into bowl. Whisk in mayonnaise, Dijon, remaining one-third of garlic, remaining ½ teaspoon sugar, ¼ teaspoon salt, and ⅛ teaspoon pepper. Whisking constantly, slowly drizzle in remaining ¼ cup oil until emulsified. Stir in mint and season with salt and pepper to taste. Slice each breast on bias ¼ inch thick. Serve, drizzled with citrus sauce.

Key Step: Cover it Up