Preheat the oven to 350 degrees F.
Put the chicken breast on a baking sheet. Drizzle with olive oil and season with salt and pepper, to taste. Put in the oven and roast until cooked through, about 35 minutes. Remove the chicken from the oven to a cutting board and let cool. Once cool enough to handle, shred the chicken meat with a fork and set aside. Cook's Note: Can be done the day before.
Turn the oven temperature up to 400 degrees F.
Put a pizza stone in the oven to preheat for 1 hour, for best results. Remove the pizza dough from refrigerator and let come to room temperature, so it will be easier to work with.
Heat a medium saucepan over medium-high heat and coat the bottom with the olive oil. Once hot, add the garlic and saute until fragrant. Add the tomato paste and stir to toast. This will add depth of flavor. Stir in the tomato puree, chili powder, paprika, cumin, and oregano and bring up to a simmer. Cook for 10 minutes, stirring occasionally, for the flavors to marry together. Season, to taste, with salt and pepper.
Thinly coat a large saute pan with olive oil and heat over a medium-high heat. When the oil is hot, add the peppers, onion and garlic. Saute until the onions are translucent, about 4 to 5 minutes, stirring occasionally. Season with salt and pepper, to taste. Remove and set aside.
Flour the counter top and roll out the pizza dough to a 13-inch diameter. Carefully remove the pizza stone from the oven and put the dough on top. Ladle on the tomato sauce to lightly cover the bottom of the pizza. Sprinkle with the chicken, vegetables, and the cheeses.
Bake the pizza until crisp and golden, about 20 minutes. Remove from the oven, then slice and serve.