How to Make It
Place chicken thighs between 2 pieces of plastic wrap. Using a meat mallet, pound chicken to an even 1/8-inch thickness. Stack pounded thighs onto each other to create 1 or 2 even rectangular stacks. Using a sharp knife, cut stacks crosswise into thirds. Place chicken pieces in a gallon-size ziplock plastic bag.
Process 1/2 cup oil, garlic, lemon juice, tomato paste, cumin seeds, paprika, salt, black pepper, and cayenne in a blender until smooth, about 30 seconds. Measure 1/4 cup mixture into a small bowl; cover and reserve in refrigerator for mushrooms. Add remaining 3/4 cup mixture to chicken in bag; seal bag, and gently massage chicken to distribute marinade. Place sealed bag in refrigerator; let marinate at least 1 hour or up to 12 hours.
Meanwhile, cut onion half lengthwise into 2 wedges; remove and discard root ends. Separate wedges into 4 large petals each. Set petals aside.
Remove chicken from bag; discard marinade. Arrange chicken pieces on a rimmed baking sheet in 3 even stacks, rotating chicken pieces as needed to keep stacks as neat and cylindrical as possible. Top each stack with 2 onion petals.
Into the center of one stack, insert 3 (12-inch) metal or soaked wooden skewers 1 inch apart, forming a triangle. Use fingers to slide chicken up skewers. Position skewers over second chicken stack; skewer stack. Repeat with remaining stack. Add remaining 2 onion petals to end of shawarma. Lay skewered chicken on a baking sheet. Press meat and onions together to form a compact, 8- to 9-inch-long cylindrical roast.
Open bottom vent of a charcoal grill completely. Light charcoal chimney starter filled with briquettes. When briquettes are covered with gray ash, pour them onto bottom grate of grill, and push to one side of grill.Place grate over coals, and lightly oil. Cover grill, and open holes of lid vents; preheat grill 10 minutes. Place shawarma on oiled grates over direct heat. Grill, uncovered, turning often, until outside of shawarma is crisp and charred, 8 to 10 minutes. Move shawarma to side of grill without coals. Grill, covered (the open vent holes should be above the chicken), turning often, until a thermometer inserted in thickest portion of chicken registers 155°F, 20 to 30 minutes. Remove shawarma from grill; tent loosely with aluminum foil, and let rest about 15 minutes. (Internal temperature of chicken will continue to rise to 165°F.)
While chicken cooks, gently toss together mushrooms and reserved 1/4 cup marinade in a large bowl. Let marinate at room temperature 15 minutes. Thread marinated mushrooms evenly onto 3 (12-inch) metal or soaked wooden skewers, threading through mushroom stems when possible to hold firmly in place. Brush kebabs all over with remaining 2 tablespoons oil.
While chicken rests, place mushroom kebabs on oiled grates over direct heat. Grill, uncovered, turning occasionally, until edges are crisp, about 10 minutes. Move kebabs to side of grill without coals. Grill, covered, until tender, about 5 minutes. Remove from grill.
Hold chicken shawarma upright on a cutting board. Using a large knife, carve downward from top of stack, cutting meat into thin strips and rotating shawarma with each cut.
Serve sliced chicken and grilled mushrooms with pitas, tomato, cucumber, parsley, hot sauce, lemon-garlic tahini sauce, and lemon wedges.