Prepare Homemade Pizza Crust.
In a large nonstick skillet, cook onion in 2 teaspoons of the hot oil over medium-high heat about 3 minutes or until crisp-tender; remove onions from skillet with a slotted spoon. Add 1 teaspoon oil, the chicken and garlic to skillet; cook and stir until chicken is no longer pink, about 4 minutes. Stir in barbecue sauce. Heat through. Transfer to a bowl; cover and keep warm. Add banana pepper rings to onion; set aside.
Remove pizza dough circles from freezer, discarding waxed paper and brush tops with some olive oil.
For a charcoal grill, carefully slide 2 of the pizza dough circles, oiled side down, onto a lightly oiled rack of an uncovered grill directly over medium-hot coals. Grill for 1 to 2 minutes or until dough is puffed in some places and starting to become firm. Carefully turn the crusts with tongs, transferring them from rack to the back of a baking sheet. Working quickly and carefully, brush oil over grilled top of crust.
For each pizza, sprinkle with 1/2 cup of the smoked mozzarella and 1/4 cup of the Asiago. Top with 1/4 of the chicken mixture and 1/4 of the onion mixture. Transfer the pizzas from the baking sheet to the rack. Grill for 1 to 1-1/2 minutes more or until cheese is melted and crust is crisp. Remove pizzas from grill. (For a gas grill, preheat grill. Reduce heat to medium-hot. Place dough circles on grill rack over heat. Cover and grill as above.)
Repeat grilling with remaining 2 dough parts and toppings. Transfer pizzas to a cutting board. Top with basil. Serve immediately. Makes four 10- to 12-inch pizzas (6 to 8 servings).
You may substitute four 10- to 12-inch packaged prebaked pizza crusts.