Preheat one zone of your Blackstone Griddle to medium-high heat and another to low/medium-low heat.
Season the chicken cutlets with olive oil, kosher salt, coarse black pepper, and Blackstone Sweet Teriyaki Seasoning. Cook the chicken on the hot zone until done, flipping as needed. Let the chicken rest for 5 minutes, then chop it into small pieces.
In a small mixing bowl, combine the cabbage, carrots, ponzu sauce, sesame oil, lime juice, rice vinegar, garlic powder, and Sweet Teriyaki Seasoning. Stir and set aside. For best results, make the slaw ahead of time to let the flavors meld.
In another bowl, mix Duke's Mayo, hoisin sauce, Korean Sweet Fire Sauce, Sriracha, rice vinegar, sesame oil, garlic powder, and lime juice. Adjust the consistency with additional lime juice or rice vinegar if needed. Set aside.
Lightly oil the griddle on the low-heat zone. Toast the flatbreads, presentation-side down, until golden brown. Flip them to toast the bottom, then remove and set aside.
On the high-heat zone, add olive oil and toss the Shishito peppers in kosher salt and pepper. Cook until blistered, tossing occasionally. Squeeze lime juice over the peppers and season with garlic powder. Remove the peppers, trim the stems, and set them aside.
Place a metal cooling rack on the high-heat zone of the griddle and arrange the toasted flatbreads on the rack. Spread a generous layer of the sauce on each flatbread, then top with chopped chicken, whole or sliced Shishito peppers, and drained slaw. Sprinkle shredded mozzarella cheese on top.
Cover the flatbreads with a Blackstone XL dome or a large aluminum pan. Spray water around the rack to create steam. Check after a few minutes, adding more water if needed, until the cheese is fully melted.
Remove the flatbreads from the rack and garnish with chopped green onions, salted cashews, and sesame seeds. Drizzle with additional Korean Sweet Fire Sauce or the prepared sauce. Slice and serve immediately.
For best results, prepare the slaw and sauce ahead of time. Keep an eye on the flatbread while toasting to avoid burning. Use the dome method to melt the cheese without drying out the toppings.