Arrange the naan on a baking sheet and pre-bake for 5 minutes. Remove from the oven and set aside.
While the naan bakes, heat the vegetable oil in a large pan over medium-high heat. Add the chicken and sauté until lightly browned, 3 to 4 minutes. Add the garlic and continue cooking until the chicken is cooked through, 1 to 2 minutes longer. Season to taste with salt and pepper and remove from the heat. Drain the excess fat and juices and set aside.
Prepare the sauce by combining the peanut butter, lime juice, sugar, soy sauce and sriracha in a small bowl. Add water in 1 tablespoon increments until the mixture reaches the consistency of thick syrup and set aside.
Distribute about 1/2 cup of the cheese between the 4 flatbreads, then top each with a portion of chicken. Top with shredded carrot, scallions and cilantro, then drizzle with the peanut sauce. Finish by topping with the remaining cheese.
Bake for 5 to 6 minutes, or until the cheese is melted and bubbly. Serve immediately.