Part of the joy of this easy-to-make weeknight meal is that you get to repurpose leftovers—make the French Onion Roast Chicken the night before, and you'll have enough delicious meat to make both this, and Chicken and Herb Crêpes the night after.
Preheat oven to 500°F. Place a pizza stone in oven while it preheats.
Place pizza dough on a lightly floured surface; roll into a 14- x 10-inch rectangle. Transfer to parchment paper. Pierce all over with a fork. Transfer dough and parchment to hot pizza stone; bake at 500°F until lightly browned, about 5 minutes.
Combine coconut milk and next 6 ingredients (through ginger) in a bowl. Remove crust on parchment from oven. Spread 1/2 cup coconut milk mixture on crust, leaving a 1/2-inch border. Top with chicken, carrots, and cheese. Return pizza on parchment to hot pizza stone in oven; bake at 500°F until cheese melts, about 4 minutes. Drizzle with remaining coconut milk mixture; top with scallions, sprouts, and cilantro. Cut into 8 pieces.