Mix the marinade: Whisk the olive oil, 3 tablespoons lemon juice, minced garlic, cumin, coriander, smoked paprika, cardamom, cinnamon, cayenne, 1 1/2 teaspoons salt, and black pepper together in a large bowl until it looks like a loose rust-colored paste.
Marinate the chicken: Pat the thighs dry with paper towels, add them to the bowl, and turn them until every piece is coated. Cover and refrigerate at least 2 hours and up to 24. Overnight gives you the deepest flavor.
Heat the oven: Pull the chicken out of the fridge about 20 minutes before it goes in. Heat the oven to 425°F and line a rimmed sheet pan with foil.
Roast: Lay the thighs on the pan in a single layer with a little space between each one so they roast instead of steam. Roast for 25 to 30 minutes, until the chicken is browned on top and a thermometer in the thickest piece reads 165°F.
Broil for the char: Switch the oven to broil and move the pan to the upper third. Broil 3 to 4 minutes, until the edges are dark and blistered. Stay at the oven door for this part, it goes from charred to burnt in under a minute.
Rest and slice: Let the chicken rest on the pan for 5 minutes, then slice it into thin strips across the grain. Toss the slices in the juices left on the foil, because that's where half the flavor went.
Make the sauce and build: Stir the Greek yogurt, tahini, 1 tablespoon lemon juice, grated garlic, and 1/4 teaspoon salt together, thinning with a tablespoon or two of water until it drizzles. Warm the pitas, pile in the chicken, cherry tomatoes, cucumber, and red onion, then spoon the sauce over and finish with parsley.
Notes
Marinate the chicken at least 2 hours and up to 24 hours; the spices only sit on the surface with anything shorter. Roast at 425°F for 25-30 minutes until the thickest thigh reads an internal temperature of 165°F, then broil 3-4 minutes for the charred edges and rest 5 minutes before slicing across the grain. Total time does not include marinating.