1Preheat oven to 475°F. Place pizza stone inside and heat for at least 15 minutes.
2Slice onion into thin strips. Heat olive oil in a cast iron skillet over medium-high heat and cook onions, stirring occasionally, until golden and soft, about 8 minutes. Set aside.
3Shred rotisserie chicken by hand, removing skin and bones. Toss diced chicken with barbecue sauce to coat well.
4Stretch store-bought dough on parchment paper into a 12-inch round. Transfer to the pizza stone.
5Spread a thin layer of barbecue sauce on the dough, leaving a half-inch border.
6Scatter caramelized onions and BBQ chicken over the sauce. Top with cheddar and mozzarella, leaving some space at the edges.
7Bake for 12–14 minutes until crust is golden and cheese is melted and bubbling.
8Sprinkle with fresh cilantro and red pepper flakes. Slice and serve hot.