Thread the onion and pepper onto the skewers. Brush with oil and season with salt and pepper.
Grill until tender, about 5 minutes per side. Remove the vegetables from the skewers and place in a bowl. Let cool slightly.
Clean the grate.
Add the chicken and barbecue sauce to the onions and peppers and stir to combine. Adjust the seasoning.
Divide the pizza dough into 4 equal pieces. On a floured surface, roll out each piece into a disc about 20 cm (8 inches) in diameter.
Place 1/4 of the chicken mixture on half of each disc, leaving a 1.5-cm (1/2-inch) border around the outside edge. Moisten the border with a little water. Fold the dough over the filling and press to seal.
Set the grill burners to low.
Grill the calzones with the lid closed for about 3 minutes per side.