Cut the chicken into 1" pieces. Heat the oil in a skillet over medium-high. Add the chicken and sauté until the edges begin to brown, about 10 minutes., Add the garlic and about half the onions, and sauté, stirring frequently, until the onions begin to soften, about 5 minutes over high heat., Add 1/2 cup of the barbecue sauce and toss to coat, continuing to cook until the sauce has thickened. If you use a thick sauce to begin with, this will happen immediately; if the sauce is thinner, it could take 3 or 4 minutes. Set aside to cool., Lightly grease two 12" or 14" round pizza pans; or, for thicker, Sicilian-style pizza, an 18" x 13" rimmed half-sheet pan., Divide your risen pizza dough in half (for 2 round pizzas); or leave it whole for one thick-crust pizza. Place the dough in the pan(s), gently stretching it to fit. If the dough fights back, cover it and walk away for 10 minutes; this will give the gluten a chance to relax. When you return, the dough should be easier to work with., Cover the dough, and let it rise for 30 to 90 minutes, depending on how thick you want your crust(s). Towards the end of the rising time, preheat the oven to 425°F., Bake the untopped crust(s) for 12 minutes. Remove them from the oven. You can either cool the crust(s) and store for a couple of days, tightly wrapped, at room temperature; or continue with the pizza., To finish the pizzas, spread about ½ cup of sauce over each round crust; or spread about 1 cup sauce on the large rectangular crust. Sprinkle the shredded cheese atop the sauce., Top with the cooked chicken mixture, warming briefly in the microwave first if it's been refrigerated. For a pleasantly crisp touch, spread the uncooked onions over the chicken mixture. If you prefer onions that are completely cooked, sauté them all with the chicken., Bake the pizzas for about 12 minutes, until the crust(s) are browned and the topping is bubbling., Remove from the oven, and serve hot or warm. Garnish with snipped fresh parsley, if desired.