Preheat the oven to 400°F (200°C) and line a baking sheet with foil; set it aside.
Make the coconut crema: whisk together 1/3 cup coconut yogurt, 1 1/2 tablespoons mayonnaise, 1/2 teaspoon honey, and 1/2 teaspoon lime juice until smooth; refrigerate until ready to use.
Prepare the peanut sauce (without peanut butter yet): in a bowl whisk 1 tablespoon reduced-sodium soy sauce, 2 tablespoons water, 1 tablespoon lime juice, 1 teaspoon fish sauce, 2 tablespoons brown sugar, 1/2 teaspoon sriracha (if using), 1/2 teaspoon garlic powder, 1/2 teaspoon dried basil, 1/4 teaspoon ground ginger, 1/4 teaspoon black pepper, and 1/4 teaspoon salt until combined; set aside.
Heat 1 teaspoon olive oil in a large nonstick skillet over high heat. Add the chopped chicken and cook, stirring, until the pieces are no longer pink on the outside (they do not need to be fully cooked through yet).
Add the prepared peanut sauce to the chicken and simmer for 2 minutes, then stir in 1/2 cup peanut butter until the sauce is smooth and the chicken is well coated.
Place the 2 naan flatbreads on the prepared baking sheet. Divide the chicken mixture evenly between the naan and spread almost to the edges.
Top each flatbread with the shredded mozzarella, thinly sliced red onion, white bean sprouts, and matchstick carrots, dividing these toppings evenly.
Bake in the preheated oven for 10 minutes, then switch to broil and broil briefly until the cheese is golden and bubbly—watch closely to avoid burning.
Remove from oven and garnish with chopped roasted peanuts, chopped cilantro, and dollops or a drizzle of the chilled coconut crema; slice and serve.