Preheat the oven to 475°F (245°C) and position a rack in the center.
Make the peanut sauce: combine peanut butter, rice vinegar, soy sauce, olive oil, crushed red pepper flakes, grated ginger, honey, and sesame oil in a food processor or blender.
Blend, adding 2 to 3 tablespoons of water as needed, until the sauce is smooth and spreadable in consistency; set aside.
Place the rolled-out pizza dough on a baking sheet or pizza stone and spread the peanut sauce evenly over the surface, leaving a small border for the crust.
Scatter the shredded chicken evenly over the sauce, then sprinkle with chopped onion and drained shredded carrot.
Top with the shredded mozzarella, distributing it evenly.
Bake the pizza in the preheated oven for 7 to 12 minutes, or until the crust is cooked through and the cheese is bubbly and lightly browned.
Remove from the oven and sprinkle with chopped fresh cilantro before slicing and serving.