On a floured pizza stone or rimless baking sheet, form pizza dough into desired shape, stretching out from center outwards. Let rest 10 minutes and stretch again. No need to make a rim, just keep dough flat as you are stretching it.
Preheat the grill to high, then grease grates with nonstick spray or with a brush and oil.
Place dough directly onto grill, close lid, and let cook 5 minutes.
Take dough off grill and flip over so that grilled size is up.
To grilled side, add 1/2 cup bbq sauce and brush it around, then add half of each of the cheeses.
In a small bowl, add remainder of bbq sauce and chicken, mix until well combined. Then add the chicken onto the pizza.
Layer red onion slices, then top with remaining cheese.
Place pizza back on the grill for another 4-5 minutes, or until crust has cooked and cheese has melted.
Pull off the grill and let rest for a couple of minutes before cutting into.