To make Spicy Peanut Sauce: Combine sauce ingredients in a small pan over medium heat. Bring the sauce to a boil. Boil gently for one minute. Divide into 2 portions for use on chicken and pizza. Set aside.
To make Thai Chicken: Cook the chicken in olive oil over medium-high heat, stirring, until just cooked, 5 to 6 minutes. Do not overcook. Set aside in the refrigerator until chilled through. Once chilled, coat the chicken with 1/4 cup Spicy Peanut Sauce. Set aside in the refrigerator.
To make the pizza: Place the pizza stone in the center of the oven and preheat to 500 degrees for 1 hour before cooking the pizzas. Use a large spoon to spread 1/4 cup spicy peanut sauce evenly over the surface of the prepared dough within the rim. Cover the sauce with 3/4 cup shredded mozzarella. Distribute half the chicken pieces evenly over the cheese followed by half the green onions, bean sprouts, and carrots, respectively. Sprinkle an additional 1/4 cup mozzarella over the other toppings and top the pizza with 1 tablespoon chopped roasted peanuts. Transfer the pizza to the oven. Bake until the crust is crisp and golden and the cheese at the center is bubbly, 9 to 10 minutes. When cooked through, carefully remove the pizza from the oven. Sprinkle 1 tablespoon chopped fresh cilantro over the hot, cheesy surface. Slice and serve. Repeat with remaining ingredients for a second pizza. (The 2 pizzas may be prepared simultaneously if you are careful in placing the pizzas at opposite corners of your pizza stone.)