Preheat oven to 450 degrees. Press or roll out crust and place on a well-oiled baking sheet. Bake for 8-10 minutes, until puffed and the bottom is lightly crisped.
Meanwhile, whisk together the peanut butter, lime juice, soy sauce, vinegar, garlic, water, and red pepper flakes (to taste).
Heat the oil in a skillet over medium-high heat. Cook the chicken until cooked through. Add the cooked chicken to a bowl and combine with the green pepper, carrot, and 1 heaping tablespoon of the sauce.
Spread the remainder of the sauce on the prepared crust. Spoon the chicken mixture evenly over the crust, sprinkle with the scallions, reserving some for garnish if desired, and sprinkle with the cheese.
Bake for 10-12 more minutes until the cheese is melted and the crust is nicely browned. Allow to cool for about 5 minutes before cutting.