Preheat a large, 2 burner cast iron grill over medium-high heat. Preheat oven to 375°.
Toss chicken in 3/4 cups of the barbecue sauce.
Divide the dough in half. On a lightly floured surface roll 1 piece into a 10-inch circle. Stretch it by hand to make it an oval so it will fit on the grill. It doesn't have to be perfectly shaped but try to make the thickness as even as possible.
Lightly brush 1 side of the dough with olive oil. Transfer to the preheated grill, oiled side down and brush the other side with olive oil. Within 1 minute the dough should begin to puff and get crisp on the bottom. Flip the dough over and grill another minute or so until marked and crisp. Transfer the grilled pizza crust to a parchment paper lined sheet pan.
Spread 3/4 cup of barbecue sauce on top of pizza crust. Scatter chicken, the cheeses, diced peppers and green onions over crust.
*Cook's Note: Do not over top the pizza or it will be too heavy for the crust and will make it soggy.
Bake the pizza in the oven for 5 minutes or until the cheese melts. Repeat with remaining ingredients.
Cut pizzas into wedges and serve with sour cream if desired.
Basic Pizza Dough:
Measure out 1 3/4 cups warm water (it should be pleasantly warm on your wrist). Sprinkle the yeast on top of the water and allow it to activate, about 10 minutes.
Put the salt and 2 cups of the flour into a food processor. Pulse 5 times to blend. Pour in the yeast and water and pulse 5 times. Add 3 tablespoons olive oil. Add the remaining flour 1 cup at a time, pulsing and scraping the sides of the bowl until well blended. As soon as the mixture is combined, dump it out onto a well-floured board and knead for 15 turns. The dough will become smooth and elastic-like.
Place the dough in a greased bowl and turn to coat all sides. Cover with plastic wrap or a tea towel. Allow the dough to rise in a warm place for 45 minutes, until doubled in size, and then punch it down.
Divide it in half; each half will make 1 (12-inch) pizza. You can make 2 pizzas, or freeze half the dough for another time. Freeze the dough in a resealable freezer bag. To thaw, remove the bag from the freezer and place in the refrigerator for 24 hours. Roll the dough out while cold and then allow it to come to room temperature before adding toppings.