In a bowl, combine the sliced chicken with Worcestershire sauce, 1 teaspoon olive oil, ½ teaspoon salt, and black pepper. Toss well and let sit while preparing the vegetables.
Heat the remaining teaspoon of olive oil and butter in a large skillet over medium-high heat. Add the onions and bell peppers with a pinch of salt. Cook for 6–8 minutes until softened and lightly caramelized. Stir in the garlic and Italian seasoning, cooking for another 30 seconds. Transfer the vegetables to a plate.
Increase the heat to high. Add a little more oil if needed, then spread the chicken into an even layer. Cook undisturbed for 1–2 minutes before stirring. Continue cooking for another 3–4 minutes until fully cooked and lightly browned. Season with the remaining salt and pepper.
Return the cooked onions and peppers to the skillet and toss with the chicken. Divide the mixture into four portions and top each with 1–2 slices of provolone cheese. Cover the skillet with a lid or foil and cook for 30–60 seconds until the cheese melts.
Spread the toasted hoagie rolls with garlic mayo, Dijon mustard, or butter if desired. Fill each roll with the cheesy chicken mixture and serve immediately.