Place chicken in a medium saucepan; add water to cover. Bring to a boil; cover, reduce heat, and simmer 20 minutes or until chicken is done. Drain and shred chicken. Set aside.
Cut chile pepper in half lengthwise; remove and discard seeds and membrane. Place pepper, skin side up, on a baking sheet; flatten with palm of hand. Broil 3 inches from heat (with electric oven door partially opened) 5 minutes or until charred. Place in ice water until cool; peel and discard skin.
Place chile pepper, 2 plum tomatoes, basil, and next 4 ingredients in container of an electric blender; cover and process until smooth, stopping once to scrape down sides. Set aside.
Coat a nonstick skillet with cooking spray; place over medium-high heat until hot. Add onion and next 5 ingredients; saute until tender. Set aside.
Unroll pizza crust, and cut into 4 equal portions; place on 2 baking sheets coated with cooking spray. Pat each portion into a 6 1/2-inch circle, forming a rim around edges. Spread pureed tomato mixture evenly over crusts. Sprinkle evenly with mozzarella and Monterey Jack cheeses. Slice remaining plum tomatoes, and arrange evenly on pizzas. Top evenly with shredded chicken, onion mixture, green onions, Parmesan cheese, and oregano. Bake at 425° for 18 minutes.