To make the marinade, whisk the olive oil, shawarma seasoning, garlic and lemon juice until well combined.
Coat the chicken with the marinade and let it sit in the fridge for at least 20 minutes, preferably a few hours. Heat a grill or griddle pan over medium heat, and grill the chicken until well cooked and browned, about 8-10 minutes. Remove from the heat and slice.
To make the tahini sauce, place all the ingredients together in a food processor and blend until well combined. You can also do this by whisk or with a fork, but it may take more time that way.
Divide the mozzarella cheese among 6 pitas and lightly toast the pita in a toaster oven, on the grill or in the oven until the cheese melts, about 2-3 minutes
Top the pita with the sliced chicken, tomatoes, banana peppers, red peppers, red onions and parsley. Drizzle the tahini sauce on top. Cut and enjoy!