Preheat the oven to 500°F (if using a baking sheet) or as high as it will go (if using a baking stone/steel; place the stone in the middle or place the steel in the bottom third of the oven before you start preheating). Let the oven preheat for about 30 minutes. Then, if you’re using a baking stone or steel, switch the oven to Broil on high.
Place a grill pan or cast-iron skillet over medium-high heat or fire up your outdoor grill. In a medium bowl, combine the chicken, ½ tablespoon olive oil, the salt, and a few grinds of black pepper. Smush everything together with your hands until both chicken thighs are evenly coated, then lay them on the grill pan or grill and cook for 3 to 4 minutes per side, just until the chicken is no longer pink in the middle. Immediate transfer the chicken to a cutting board.
Slice or chop the chicken into bite-size pieces, put them in a bowl, and mix with 2 tablespoons of barbecue sauce, making sure every piece is coated.
Stretch or roll out your dough to a 12- to 14-inch circle, then transfer it to a baking sheet / pizza pan or a lightly floured pizza peel (if using a baking stone/steel).
In a medium bowl, mix together the two different cheeses.
Brush the stretched- or rolled-out dough with the remaining ½ tablespoon of olive oil, making sure to get it all the way to the edges. Next, pour the barbecue sauce onto the dough and use the back of a spoon to spread it out evenly, leaving a ½-inch border all around. Top with the sauce-coated chicken, then the cheese, and finally the raw onion.
Transfer the pizza to the oven and bake until the crust is evenly browned on the bottom and the cheese and toppings have browned in spots—8 or 9 minutes on the baking sheet, 5 to 8 minutes on the baking stone/steel.
Take the pizza out of the oven, let it cool down for 5 minutes or so, then sprinkle with scallions, slice, and serve.