Preheat the oven to 400 F. Place a large, greased baking sheet in the oven (a hot baking sheet helps when cooking the pizza).
Make the BBQ sauce. Add all sauce ingredients to a large saucepan over high heat. Bring to a low boil and then reduce the heat to low. Let simmer for 20 minutes until the sauce has thickened, stirring occasionally. Set aside.
Using a fork, shred the chicken. Toss the shredded chicken with 1 cup of the prepared BBQ sauce and set aside.
On a floured surface, roll out the pizza dough with a rolling pin until the dough is about 1/4-inch thick. Drizzle 2 tablespoons of olive oil on top of the dough, followed by 1/2 cup of the prepared BBQ sauce, and spread evenly.
Add the shredded BBQ chicken on top, followed by the shredded mozzarella cheese, gouda cheese, and red onion. Drizzle with remaining 1 tablespoon of olive oil.
Bake in the preheated oven for 12-15 minutes, or until the crust is cooked through and the cheese is melted. Broil for a few minutes until the cheese begins to brown, watching carefully that it doesn’t burn.
Remove pizza from the oven, let cool, and cut into slices.