Grate 2 teaspoon zest from the lemons, then halve the lemons crosswise and set aside. In a small bowl, stir together the zest, oil, garlic, oregano, 1¼ teaspoons salt and ½ teaspoon pepper. Transfer ¼ cup of the mixture to a large bowl, add the chicken and turn to coat. Marinate at room temperature while preparing the grill.

Prepare a charcoal or gas grill for indirect cooking. For a charcoal grill, ignite a large chimney of coals, let burn until lightly ashed over, then distribute the coals evenly over one side of the grill bed and open the bottom grill vents. Heat the grill, covered, for 5 minutes, then clean and oil the cooking grate. For a gas grill, turn all burners to high and heat, covered, for 15 minutes, then clean and oil the cooking grate; leave the primary burner on high and turn the remaining burner(s) to low.

Place the chicken skin side up on the cooler side of the grill. Cover and cook for 15 minutes. Using tongs, reposition the chicken so that the pieces furthest from the fire are now closest; keep the chicken skin-side up. Re-cover and continue to cook until the thickest part of the breast, if using, reaches 160°F or the thickest part of the thighs and drumsticks, if using, reach 175°F, another 5 to 10 minutes.

Using tongs, flip the chicken skin-side down onto the hot side of the grill. Cook until the skin is lightly browned and crisp, 1 to 2 minutes, moving the chicken as needed to avoid flare-ups. Transfer the chicken skin side up to a platter and let rest for about 5 minutes.

While the chicken rests, cut each lemon in half and grill the lemon halves cut side down on the hot side of the grate until grill-marked, 2 to 3 minutes. Squeeze 2½ tablespoons juice from 1 or 2 of the grilled lemon halves, then stir the juice along with the parsley into the reserved garlic-lemon oil to make the salmoriglio. Serve the sauce and the remaining grilled lemon halves with the chicken.

Follow the recipe to marinate the chicken. Heat the oven to 450°F with a rack in the middle position. Place the chicken skin side up and the lemon halves cut side up (trim off their pointed end so they sit flat) on a rimmed baking sheet. Roast until the chicken is light golden brown and the thickest part of the breast, if using, reaches 160°F or the thickest part of the thighs and drumsticks, if using, reaches 175°F, 30 to 40 minutes. Using tongs, transfer the chicken skin side up to a platter and let rest for 5 minutes. While the chicken rests, follow the directions above for making the salmoriglio using the roasted lemon halves.