In a small saucepan, melt 1 tablespoon of butter over medium-low heat. Whisk in the flour and cook until the mixture is light brown, 2 to 3 minutes.
Slowly whisk in the milk and bring the mixture to a boil. Remove it from the heat and add ¼ cup of the cheese. Mix until the cheese is melted, adding more as needed to reach desired consistency. Season with salt and pepper to taste. Keep the cheese sauce warm and set it aside.
Melt 1 tablespoon of butter in a large skillet over medium-high heat. Add the bell pepper, onion, and mushrooms and sauté until they're tender and beginning to caramelize, 6 to 8 minutes. Season with salt and pepper to taste. When they're cooked, remove the vegetables from the heat and transfer them to a bowl.
Return the pan to medium-high heat and melt the remaining 1 tablespoon of butter. Add the sliced chicken and season with the Italian seasoning, as well as salt and pepper to taste. Cook, flipping halfway through, for about 3 to 4 minutes or until the internal temperature is 165 F. Return the vegetables to the pan to incorporate.
Heat the hoagie rolls under the broiler for 2 to 3 minutes. Spread ½ tablespoon of mayonnaise on each roll, optionally. Split the chicken and vegetables evenly among the 4 rolls, and top with the provolone cheese sauce. Serve immediately.