In a large, heavy skillet, heat the oil over medium heat. Add the onion and cook, stirring constantly, until softened and brown, 15 to 20 minutes. Season with salt and pepper and remove from the heat.
Invert a large baking sheet and place it on a rack in the lower third of the oven; preheat the oven to 500 degrees . Place a second inverted baking sheet on a work surface and cover with parchment paper. Stretch the pizza dough to fit the parchment, then brush with oil.
In a small bowl, combine the barbecue and tomato sauces. Spread the sauce over the dough, leaving a 3/4-inch border. Top with the cheese, shredded meat, bell peppers and caramelized onions.
Slide the pizza and parchment paper onto the preheated baking sheet and bake until the crust is golden and the cheese is bubbly, about 20 minutes.