Make the coconut crema: whisk together the coconut yogurt, 1 1/2 tablespoons mayonnaise, 1/2 teaspoon honey, and 1/2 teaspoon lime juice until smooth; refrigerate until serving.
Preheat the oven to 400°F (200°C) and line a baking sheet with foil or a nonstick mat.
Whisk the peanut sauce base: in a medium bowl combine 1 tablespoon reduced-sodium soy sauce, 2 tablespoons water, 2 tablespoons brown sugar, 1 tablespoon lime juice, 1 teaspoon fish sauce, 1/2 teaspoon Sriracha (if using), garlic powder, dried basil, ground ginger, salt, and pepper; set aside.
Heat 1 teaspoon oil in a large nonstick skillet over medium-high heat. Add the chopped chicken and sauté until no longer pink on the outside, about 3–5 minutes.
Add the prepared peanut sauce (from the bowl) to the skillet and simmer with the chicken for about 2 minutes, then stir in 1/2 cup peanut butter until the sauce is smooth and the chicken is evenly coated.
Assemble the flatbreads: divide the chicken-peanut mixture evenly between the 2 naan, spreading it almost to the edges; top each with shredded mozzarella, thinly sliced red onion, matchstick carrots, and bean sprouts.
Bake on the prepared sheet for 10 minutes, then broil briefly if desired until the cheese is golden and bubbly.
Garnish each flatbread with chopped peanuts, chopped cilantro, and dollops or a drizzle of the chilled coconut crema; slice and serve.