Whisk the tahini, lemon juice, garlic and salt in a small bowl. Gradually whisk in the water until smooth and creamy. Add more water, 1 tablespoon at a time, if needed to reach a pourable consistency.
Place a pita directly on a rack over a medium gas flame. Cook until lightly charred, about 30 seconds per side. Cut into 4 triangles.
Heat the olive oil in a large skillet over medium-high heat. Add the onions and cook, stirring occasionally, until deeply golden, 8 to 10 minutes.
Add the chicken and cook, stirring occasionally, until lightly browned and cooked through, about 5 minutes. Add the shawarma seasoning, cumin, chicken grill seasoning, pepper and salt, and toss to coat.
Spread the tahini over one side of each serving plate and arrange the chicken on the other. Garnish with parsley and sliced hot pepper. Serve with the charred pita.