Whisk together mayonnaise, white vinegar, lemon juice, garlic powder, onion powder, smoked paprika, black pepper, cayenne (optional), and sugar. Adjust salt and sugar to taste.
Shred rotisserie chicken and toss with ¼ to ½ cup of the sauce. Save the rest for serving.
Preheat broiler. Slice rolls lengthwise, leaving a hinge. Fill evenly with sauced chicken and top with shredded cheese.
Broil subs for 2–4 minutes until cheese melts and rolls toast lightly. Watch closely.
Top with pickles, slaw, onions, or jalapeños. Drizzle extra sauce if desired.
Serve warm with extra sauce. Pair with chips, fries, or salad.