Position rack in center of oven. Place large baking sheet on rack and preheat oven to 450°F. for 30 minutes.
Heat olive oil in heavy medium skillet over medium-high heat. Season chicken with salt and pepper. Add chicken to skillet and sauté until just cooked through, about 5 minutes per side. Transfer chicken to plate; let rest 5 minutes.Cut chicken crosswise into 1/3 inch-wide slices. Using slotted spoon, transfer chicken to medium bowl.Toss with 1/4 cup barbecue sauce. Spread half of cheese on Boboli. Arrange chicken slices on Boboli, spacing evenly. Spoon any remaining barbecue sauce from bowl over. Sprinkle red onion over chicken. Drizzle with remaining 1/4 cup barbecue sauce. Sprinkle remaining cheese and green onion over.
Transfer pizza to hot baking sheet. Bake until bottom of crust is crisp and cheese on top melts, about 14 minutes. Let pizza stand 5 minutes.