Grilled BBQ Pizza

Name: Grilled BBQ Pizza
Ingredients: 6 tablespoons olive oil, divided; 1 medium onion, sliced 1/4-inch thick; 1 medium green pepper, seeded and sliced in 1/2-inch strips; 2 medium garlic cloves, smashed; Salt and freshly ground black pepper; Store-bought pizza dough, divided into 4 pieces; Flour, for rolling; 1/4 cup BBQ sauce (recommended: Neely's BBQ Sauce, recipe follows); 2 cups cooked pulled pork, optional; 8 ounces shredded mozzarella; 1/4 cup grated Parmesan; 2 cups ketchup; 1 cup water; 1/2 cup apple cider vinegar; 5 tablespoons light brown sugar; 5 tablespoons sugar; 1/2 tablespoon fresh ground black pepper; 1/2 tablespoon onion powder; 1/2 tablespoon ground mustard; 1 tablespoon lemon juice; 1 tablespoon Worcestershire sauce