Preheat oven to 475°F (245°C). If using a baking sheet, lightly oil it or line with parchment.
Stretch or roll the pizza dough into a roughly 12-inch circle on a floured surface, then transfer to the prepared pan.
Toss the cooked chicken with 1/4 to 1/2 cup buffalo sauce until evenly coated, reserving a small amount of sauce for drizzling if desired.
Brush the dough edge with olive oil and season with a pinch of salt and pepper.
Spread remaining buffalo sauce evenly over the dough, leaving the crust exposed, then sprinkle 1 1/2 cups of shredded mozzarella over the sauce.
Distribute the buffalo-coated chicken evenly over the cheese, then top with the remaining mozzarella and Parmesan if using.
Bake in the preheated oven for 12–18 minutes, until the crust is golden and the cheese is bubbling and slightly browned.
Remove pizza from oven and let rest 2–3 minutes, then drizzle with ranch or blue cheese dressing and sprinkle sliced green onions before slicing to serve.