Make the marinade: In a large bowl, combine the paprika, cumin, coriander, ginger, turmeric, allspice, cinnamon, 2 teaspoons salt, and½ teaspoon pepper.

Add the olive oil and whisk until smooth. Add the yogurt, onion, and garlic and mix well. Set the marinade aside.

Pat the chicken pieces dry with a paper towel. If using breasts, cut the breasts i11 half crosswise, so you have one thicker, rounded piece and one thinner, triangular piece. Working in batches, place the chicken pieces on a large cutting board and cover them with a sheet of parchment paper. Using a meat mallet or rolling pin, pound the chicken through the paper until each piece is about the same thickness as the others, between½ and¾ inch thick. (Be careful not to pound so hard you tear the chicken.)

Add the chicken pieces to the marinade and toss well. Cover and refrigerate for at least 3 hours or up to 24.

Meanwhile, make the tahini-yogurt sauce: In a small bowl, combine the yogurt, tahini, lemon juice, garlic, and salt. Mix well. Refrigerate for at least 30 minutes (and up to 3 days) until ready to use.

Before grilling, let the chicken come to room temperature for 30 minutes. If using a charcoal or gas grill, clean the grates and brush them with oil. Set the grill to medium-high or heat a grill pan slicked with olive oil on the stovetop over medium­-high.

Lift the chicken from the marinade (no need to brush off excess marinade) and grill until cooked through, 6 to 8 minutes on each side, covering the grill halfway through, if necessary, to retain the heat.

Transfer the chicken to a plate, sprinkle generously with salt and squeeze the lemon half over the pieces. Allow the chicken to rest for at least 10 minutes before serving. Garnish with chopped fresh parsley or mint and serve the tahini-yogurt sauce on the side.