4 cups fresh basil leaves
¾ cup extra-virgin olive oil
5 garlic cloves, peeled
1½ tablespoons lemon juice
Salt and pepper
2 ounces Parmesan cheese, grated (1 cup)
4 (12-ounce) bone-in split chicken breasts, trimmed
Process basil, ½ cup oil, garlic, lemon juice, and ¾ teaspoon salt in food processor until smooth, about 1 minute, scraping down bowl as needed. Transfer ¼ cup pesto to large bowl and set aside for marinade. Add Parmesan to pesto left in processor and pulse to incorporate, about 3 pulses; transfer ¼ cup pesto to small bowl and set aside for stuffing. Add remaining ¼ cup oil to pesto left in processor and pulse to incorporate, about 3 pulses; set aside for serving.
Starting on thick side of breast, closest to breastbone, cut horizontal pocket in each breast, stopping ½ inch from edge so halves remain attached. Season chicken, inside and out, with salt and pepper. Place 1 tablespoon pesto reserved for stuffing in each pocket. Tie each chicken breast with 2 pieces kitchen twine to secure. In large bowl, rub chicken with pesto reserved for marinade, cover, and refrigerate for 1 hour.
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. Turn all burners to medium-low. (Adjust burners as needed to maintain grill temperature of 350 degrees.)