Preheat the Blackstone to medium-high heat for cooking the steak and vegetables, leaving a medium-heat zone for toasting the wraps.
In a bowl, mix the mayo, A1 steak sauce, horseradish, Worcestershire, garlic powder, salt, black pepper, and Cheesesteak Seasoning until smooth. Set aside or refrigerate until ready to use.
Add a light drizzle of oil to the griddle, then add the shaved ribeye, diced bell peppers, and diced onion all at once. Season with salt, black pepper, and optional extra Cheesesteak Seasoning. Cook while chopping and tossing until the beef is browned and the vegetables are tender, about 6–8 minutes.
Shape the cooked steak and vegetables into a mound or long line. Place the slices of white American cheese directly on top, cover with a melting dome, and add a small splash of water to steam until the cheese is fully melted. Toss everything together until evenly combined and cheesy.
Transfer the cheesesteak mixture to a bowl. Lay the tortillas flat and spread a generous layer of the special sauce down the center of each tortilla. Spoon in a hearty portion of the cheesesteak mixture and roll tightly, tucking in the sides as you go.
Lightly butter or oil the griddle and place the wraps seam-side down over medium heat. Toast for 1–2 minutes per side until golden and lightly crisp.
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