Grilled Chicken with Rosemary White Barbecue Sauce

To make the sauce, whisk the mayonnaise, lemon zest and juice, rosemary, and garlic together. (The sauce can be made up to 3 days ahead, covered, and refrigerated.) Build a fire for indirect grilling in an outdoor grill. For a charcoal grill, leave the coals heaped in a mound in the center. For a gas grill, preheat on high. Then leave one burner on high and turn the other burner(s) off. Season the chicken with the salt and pepper. Lightly oil the cooking grate. Place the chicken on the unheated areas of the grill (that is, on the perimeter around, but not over, the coals in the charcoal grill, or over the off burners of the gas grill). Cover and grill for 20 minutes. Turn the chicken, cover, and continue grilling until an instant-read thermometer inserted in the thickest part of the breast reads 165°F, about 20 minutes more. Move the chicken to the hot area of the grill (over the coals of the charcoal grill, and over the high burner of the gas grill). Brush the chicken all over with the mayonnaise mixture. Grill, turning and basting occasionally with the remaining mayonnaise mixture, until the mayonnaise has reduced to a mostly transparent glaze, 3 to 5 minutes. Transfer to a platter and serve hot, with the remaining sauce passed on the side, if you wish.