Remove giblets from chicken and trim any excess skin and fat. Place the chicken on a cutting board breast-side down. Working from the cavity to the neck, use kitchen shears or a sharp knife to cut down each side of the backbone; remove and discard (or reserve to make stock). Turn the chicken cut-side down and flatten the breastbone with the heel of your hand.

Place the chicken in a 9-by-13-inch glass baking dish. Whisk wine (or port), vinegar and mustard in a small bowl. Pour the marinade over the chicken and massage it in. Marinate the chicken, breast-side down, for at least 1 hour and up to 24 hours.

Preheat a gas grill to medium-high, then turn off one burner (leaving 1 to 2 burners lit, depending on your grill). Or build a two-zone fire in a charcoal grill (coals on one side) and let it burn down to medium-high heat (450 degrees F).

Oil the grill rack. Remove the chicken from the marinade (reserve the marinade) and sprinkle with 1/2 teaspoon salt. Grill the chicken, skin-side down, over the unlit portion of the grill until the skin is well-marked and browned around the edges, 25 to 30 minutes. Carefully turn it over and continue grilling until a meat thermometer inserted into the thickest part of a thigh registers 160 degrees F, 20 to 25 minutes more. Transfer the chicken to a clean cutting board and let rest for 10 minutes.

Meanwhile, toss zucchini with oil and the remaining 1/4 teaspoon salt in a large bowl. Grill until lightly charred, 3 to 4 minutes per side.

Pour the reserved marinade into a small saucepan. Bring to a boil over high heat. Cook until the sauce is reduced by half, about 8 minutes.

Carve the chicken and arrange with the zucchini on a serving platter; serve with the sauce.