Marinate chicken breasts in refrigerator in 1 cup of barbecue sauce for at least 30 minutes.
Dissolve yeast in warm water in a large mixing bowl. Add cornmeal mix, salt, and sugar, and stir to combine. Gradually add flour until the dough pulls away from the side of the bowl and is soft.
Flour your hands and turn dough onto lightly floured surface. Knead by turning and folding for 10 minutes or pulse in food processor or electric mixer.
Pour about a tablespoon of olive oil into a medium bowl. Place dough into bowl and turn once to coat with oil. Cover bowl with plastic wrap and allow to rise in warm place until doubled in size, about 1 hour. Punch down the dough. Dough may be used immediately or wrapped in plastic wrap and refrigerated for up to 24 hours.
If using refrigerated dough, allow dough to sit at room temperature to warm for 30 minutes before rolling.
Heat oven to 450 degrees. Lightly coat two pizza pans or baking sheets with olive oil or no-stick cooking spray.
Divide dough in half and make into two balls. Sprinkle pan surfaces with flour or cornmeal. Roll dough to fit and place on prepared pans. Let the dough rise in pans about 15 minutes. Bake for 12 to 20 minutes, or until crust is golden brown.
To prepare the chicken: Heat oven to 350 degrees. Place chicken in a single layer in a shallow baking dish. cover with foil. Bake until the juices run clear, about 20 minutes. Cut chicken into 1/4 to 1/2 inch pieces.
Spread 1/2 cup of remaining barbecue sauce on each pizza crust. Top with chicken and vegetables. Bake for 15 minutes.
. Remove from oven and top each pizza with 1 cup of shredded cheese. Return to oven and bake for 5 minutes, or until cheese is melted.