FOR THE TZATZIKI SAUCE: Whisk lemon juice and garlic together in small bowl. Let stand for 10 minutes. Stir in yogurt, cucumber, mint, parsley, and salt. Cover and set aside.
This tzatziki is fairly mild; if you like a more assertive flavor, double the garlic. A rasp-style grater makes quick work of turning the garlic into a paste. We like the chicken as a wrap, but you may skip the pita and serve the chicken, vegetables, and tzatziki with rice. You will need four 12-inch metal skewers. For tips on wrapping the sandwich in foil, see related content.
FOR THE CHICKEN: Dissolve 2 tablespoons salt in 1 quart cold water. Submerge chicken in brine, cover, and refrigerate for 30 minutes. While chicken is brining, combine oil, parsley, lemon zest and juice, honey, oregano, and 1/2 teaspoon pepper in medium bowl. Transfer 1/4 cup oil mixture to large bowl and set aside to toss with cooked chicken.
Remove chicken from brine and pat dry with paper towels. Toss chicken with remaining oil mixture. Thread 4 pieces of bell pepper, concave side up, onto one 12-inch metal skewer. Thread one-quarter of chicken onto skewer. Thread 2 chunks of onion onto skewer, and place skewer on plate. Repeat skewering remaining chicken and vegetables on 3 more skewers. Lightly moisten 2 pita breads with water. Sandwich 2 unmoistened pita breads between moistened pita breads and wrap stack tightly in lightly greased heavy-duty aluminum foil.
FOR A CHARCOAL GRILL: Open bottom vent completely. Light large chimney starter mounded with charcoal briquettes (7 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 skewers on hotter side of grill and cook, turning occasionally, until chicken and vegetables are well browned on all sides and chicken registers 160 degrees, 15 to 20 minutes. Using fork, push chicken and vegetables off skewers into bowl of reserved oil mixture. Stir gently, breaking up onion chunks; cover with foil and let sit for 5 minutes.
Meanwhile, place packet of pitas on cooler side of grill. Flip occasionally to heat, about 5 minutes.
Lay each warm pita on 12-inch square of foil. Spread each pita with 2 tablespoons tzatziki. Place one-quarter of chicken and vegetables in middle of each pita. Roll into cylindrical shape and serve.
MARINATE TWICE
Briefly soaking the grilled chicken and vegetables in some reserved marinade before wrapping them in a pita rehydrates their dry exteriors and delivers bright citrus punch.
SOFTENING SUPERMARKET PITA
To soften up dry, tough supermarket pita, we moisten two of the breads with a little water and then stack them on either side of two unmoistened pieces. Then we steam the breads in a foil-wrapped stack on the cooler side of the grill while the cooked chicken rests.
A VEGETABLE SHIELD
The age-old problem with grilling meat on a stick: The end pieces overcook. We protect the chicken by threading pepper and onion pieces on the ends. The charred vegetables taste great in the sandwich, too.