In a small bowl, dilute the sugar in warm water; add the yeast. When yeast begins to foam, stir in salt and oil.
In a mixing bowl, combine the flours. With electric mixer on medium speed, slowly add yeast mixture. Mix until well combined. Coat a large bowl with cooking spray. Place dough in bowl and cover. Allow dough to rise in a warm area for 1 to 1 1/2 hours.
As the dough rises, prepare the toppings. In a medium saute pan on medium heat, add the onion and oil. Cook the onions until brown and caramelized, stirring frequently.
Heat oven to 400 F. Roll out the dough ball on a floured surface to 1/2-inch or desired thickness. Place dough on baking sheet. Top with barbecue sauce, chicken, onion, peppers and cheeses. Bake for 15 to 20 minutes, or until cheese melts and crust is crisp.