Preheat the oven to broil and line a baking sheet with parchment paper or heavy-duty foil.
Use a sharp knife to slice a V along the top of each hoagie roll and remove the center, taking care not to cut through the bottom.
Divide 2½ tablespoons of the butter among the 4 rolls and spread on the inside; place rolls on the prepared sheet and broil 2–3 minutes until golden and toasted, then remove.
In a mixing bowl, toss the sliced chicken with the dried Italian seasoning, kosher salt, and cracked black pepper until evenly coated.
Heat 1 tablespoon olive oil in a 10–12-inch skillet over medium-high heat, add the seasoned chicken, and cook 4–5 minutes per side until cooked through; transfer to a bowl.
Return the skillet to medium-high heat, add the remaining 1 tablespoon olive oil and the remaining 1 tablespoon of butter; once melted, add the sliced green pepper and sliced onion and sauté 2–3 minutes, stirring frequently.
Add the sliced mushrooms and minced garlic to the skillet and continue to sauté 2–3 minutes until the onion is opaque and mushrooms are cooked through, then remove from heat.
Return the cooked chicken to the skillet with the vegetables and stir to combine.
Divide the chicken and vegetable mixture among the toasted rolls, layer provolone slices over the filling, then top with remaining filling (or place cheese on top in the skillet before adding to rolls).
Place the assembled sandwiches under the broiler 1–2 minutes until the cheese is melted, then remove and serve.
