Preheat the grill to 400-450 degrees. Brush the grates with oil. Brush both sides of the pizza crust lightly with oil.
In a saucepan, combine the peanut butter, soy sauce, brown sugar, rice vinegar, chili garlic sauce, and water. Cook over medium heat, stirring, until everything is melted together. Pour half the sauce into a bowl, add the chicken, and toss to coat.
Spread the rest of the sauce over the dough. Top with ¾ cup cheese, the chicken, and then the rest of the cheese. Grill for 8-10 minutes or until the crust is browned and the cheese has melted.