Preheat oven to 450°. (For a crispy crust, preheat a pizza stone on the lowest rack in the oven.)
Combine first 7 ingredients in a skillet over medium-high heat. Bring to a boil, reduce heat, and simmer, stirring often, 5 minutes or until thickened and reduced (Mixture should be thick like pizza sauce.)
Roll dough to 1/4-inch thickness on a floured baking sheet or pizza peel. Brush edges of crust with olive oil. Spread sauce to edges of crust; top with chicken and cheeses.
If desired, transfer pizza to pizza stone dusted with cornmeal or semolina flour; bake 10 to 12 minutes. Remove from oven; sprinkle with green onions and cilantro.