Preheat your barbecue to medium-high. Combine first 5 ingredients in a bowl. Lightly oil the bars of your grill. Grill chicken 3 to 4 minutes per side, or until cooked through. Transfer to a plate.
Clean the grill. Lightly oil the bars again. Grill each pita until nicely coloured on one side, about 1 minute. Now set on work surface, grilled side up. Turn barbecue to low. Spread the grilled side of each pita with barbecue sauce. Slice the chicken. Top each pizza with cheese, chicken, bell pepper and onion. Carefully set the pita back on the grill, cheese side up. Close the lid and cook 3 minutes, or until lightly charred on the bottom and the cheese is melted.